The Refutation of Panini Grammar about Narayana being a proper noun is beautifully given in this link with proper proofs. Read that. That will be more than enough.
I am just giving refutation from Shrutis and Smriris.
The Name Nārāyaña can have very meanings. The name can be split into Nāra + āyana.
Where, Nāra can mean water and āyana means abode. So, the one whose abode is water means Lord Vishnu.
Nāra can also mean Purusha (Jiva) and āyana as usual abode. So, it means the one whose abode is Jiva viz Atman.
“aatmaaya namaH |” (Taittiriya Aranyaka 10.16.2).
“Salutations to Him (Rudra), He who is the Spirit – Atman – dwelling in all creatures”.
"devana hridayebhyo namah”
“Salutations to Rudra who is the indweller / sparkling heart of all Gods” – Yajurveda Sri Rudram Anuvaka-9.
So, here Narayana can mean Bhagawan Shiva.
- Nāra = Purusha and āyana can also mean son. It's same as the name Dakshāyiñi means the daughter of Daksha viz Mata Sati. So, here Narayana means the 1st born of Purusha.
“yo devaanaaM prabhavashchodbhavashcha vishvaadhipo rudro maharshhiH |
hiraNyagarbha.n janayaamaasa puurva.n sa no buddhyaa shubhayaa sa.nyunaktu” (Sve Upa. 3:4).
“He, the omniscient Rudra, the creator of the gods and the bestower of their powers, the support of the universe, He (Rudra/Purusha) who, in the beginning, gave birth to Hiranyagarbha—may He endow us with clear intellect!”
So, here Narayana means the 1st born viz Lord Brahma.
- Nāra = Purusha and āyana can mean direction It's same as Rāmāyaña where āyana means to go towards to direction of Lord Rāma or to reach him as destination. So, here Narayana means the entity that goes towards the direction of Purusha to reach him as destination.
“prakR^ityashhTakaruupa.n cha sthaana.n gachchhati kuNDalii | kroDiikR^itya shiva.n yaati kroDiikR^itya viliiyate |” (YogaKundalini Upanishad 1:74).
“Kundalini assumes the eight forms of Prakriti (matter) and attains Shiva by encircling him and dissolves itself in Shiva”.
So, Here Narayana means the Kundalini Parmeshwari Maa Teipura Sundari.
- Nāra = Purusha and āyana means contained. So, here Narayana are the one who are contained in this Purusha.
All the gods are contained inside lord Shiva (Purusha) only.
“yásya tráyastriṃśad devā́ áṅge sárve samā́hitāḥ |
skambháṃ táṃ brūhi katamáḥ svid evá sáḥ |” (Atharva Veda X:7:13).
“Who out of many, tell me, is that Skambha (Agni Lingam) He in whose body are contained all three-and-thirty Deities.”
So, Narayana Shuktam says.:
In the midst of that light, the divine resides. He is Brahma, Shiva, Sustainer (Hari), and Indra. He is imperishable, Swayambhu (self-located) and divine.-(12).
The logic is same as Vāsudeva in vedas in not “VasudEva Sutam iti VAsudEva (Krishna)”. Vasu means the the tattvas which stays inside n envelope and Deva means the shining one. So, the meaning of Vasudeva is the one who stays inside everything snd shine from inside viz the Atman.
Anyways, Let us see some examples from Puranas, Shrutis, Smritis and Itihasas (Mahabharata and Ramayana) also.:
1. Lord Vishnu as Narayana.:
नारायणेति नामापि तस्यसीच्छ्रुतिसंमतम् ।।
नान्यत्किंचित्तदा ह्यासीत्प्राकृतं पुरुषं विना।।५४॥
54. “As approved in the Vedas, his name came to be established as Nārāyaṇa (Having water as abode). Excepting for that Primordial Being there was nothing then.”
(Said by Lord Brahma himself).
- Markandeya Samasya Parva in Mahabharata.:
“ In ancient times I called the waters by the name of Nara; and because the waters have ever been my ayana or home, therefore have I been called Narayana (the water-homed). O best of regenerate ones, I am Narayana, the Source of all things, the Eternal, the Unchangeable. I am the Creator of all things, and the Destroyer also of all. I am Vishnu, I am Brahma and I am Sakra, the chief of the gods. I am king Vaisravana, and I am Yama, the lord of the deceased spirits. I am Siva, I am Soma, and I am Kasyapa the lord of the created things.“
Even here you can see Narayana is applied to all Lord Vishnu, Lord Brahma, Lord Shiva, Lord Indra, etc also.
“I am called Narayana, and I am bearer of the conch-shell, the discus and the mace.”
(Said by Lord Vishnu himself).
- Mahabharata Kirat Parva.:
The heroic Arjuna, with mighty arms, is the god incarnate styled Nara. He practised austerities of old in the Vadari forest. The God Narayana (Krishna) is his friend.
(Said by Lord Shiva)
*"You are the Lord Narayana (Rama) himself the glorious god, who wields the discus. You are the Divine Boar with a single tusk, the conqueror of your past and future enemies.You are also Brahma, the imperishable, the Truth abiding in the middle as well as at the end of the universe.”$
(Said by Lord Brahma)
So, here also Narayana name applies to both Lord Brahma and Vishnu.
2. Lord Brahma as Nārāyaṇa.:
नारायणाख्यो भगवान्ब्रह्मा लोकपितामाहः ॥ १,३.३ ॥उत्पन्नः प्रोच्यते विद्वन्नित्यमेवोपचारतः ॥ १,३.४ ॥
“Hear then how the deity, the beholder of the name Náráyána ,the great parent of the world, Brahmá created all existent things.”
(Said by Rishī Parashara)
तदा समभवद्ब्रह्मा सहस्त्राक्षः सहस्त्रपात् ॥ १,६.२ ॥सहस्त्रशीर्षा पुरुषो रुक्मवर्णस्त्वतीन्द्रियः ।
ब्रह्मा नारायणाख्यस्तु सुष्वाप सलिले तदा ॥ १,६.३ ॥
इमं चोदाहरन्त्यत्र श्लोकं नारायणं प्रति ।
In that vast sheet of water there appeared Brahmā of thousand eyes and thousand feet , the Purusa of thousand heads and of golden colour . He was beyond the scope of sense - organs . Brahmå who was called Nårāyana lay asleep on the cosmic waters at that time . They cite this verse in this context , regarding Narayana , the very embodiment of Brahman ( or the Veda ) and who is the Lord and the origin of the world and the cause of dissolution of the universe !
(Said by Lord Kurma Vishnu)
- Manu Smriti 1.9–11 (Not Puranic References, But from Smriti).:
तदण्डमभवद्धैमं सहस्रांशुसमप्रभम् ।तस्मिञ्जज्ञे स्वयं ब्रह्मा सर्वलोकपितामहः ॥ ॥ आपो नारा इति प्रोक्ता आपो वै नरसूनवः । ता यदस्यायनं पूर्वं तेन नारायणः स्मृतः| यत्तत्कारणमव्यक्तं नित्यं सदसदात्मकम् । तद्विसृष्टः स पुरुषो लोके ब्रह्मेति कीर्त्यते ॥
He became a golden egg (Hiranyagarbha), in brilliancy equal to the sun ; in that ( egg ) he himself was born as Brahma , the progenitor of the whole world_(9).Water is called ' nara , ' — water being the offspring of nara ; since water was created by him (or is recided by that being) , he is , on that account , described as ' nārāyaṇa .'— ( 10 ) That which is the cause—unmanifest, eternal and partaking of the nature of the existent and the non-existent,—the being produced by that (cause) is described among people as ‘brahmā.’—(11)
So, Lord Brahma being first born on water called as Narayana.
3. Maa Lalita Tripurasura Sundari as nārāyaṇa.:
- In the Kurma Purana, when Himavan praises Devi he says.:
sahastrabāhuṃ puruṣaṃ purāṇam |
śayānamantaḥ salile tathaiva |
nārāyaṇākhyaṃ praṇato ‘smi rūpam ||” (kUrma PuraNa 1:11:245).
“I salute thy form called Narayana, O Lalita, which has a thousand heads, which is of infinite energy, having a thousand arms, the ancient Person, reclining on the waters”.
- Narayana(Vishnu) says to Virabhadra in Brahmanda Purana.:
“ekaiva shaktiH parameshvarasya | bhinnA chaturdhA viniyogakAle |
bhoge bhavAnI samareShu durgA | kopeShu kALI puruSheshu viShNuH |”
“The ancient Sakti of the Lord (Parameshwara)appears in four forms; that Sakti becomes Bhavani in her bhoga form, in battle she takes the form of Durga; in anger that of Kali; and she is present in her male form as me (Vishnu/Narayana).”
- Lalita Saharanama Strotam.:
Karanguli nakhotpanna narayana dashakrutih | Mahapashu patastragni nirdagdhasura sainika |” (Lalita Sahasranama Stotram 1:32).
“Lalita – who out of her finger nails re-created all the ten Incarnations of Narayana (Vishnu) to destroy the Asuras slain by Him in His incarnations and now recreated magically by demon-Bhanda out of a missile of his. She again burned to death the armies of demons with the fire of the great missile maha-pashupata”.
So, from above verses it is proved that Narayana us also another name of Maa Tripurasura Sundari.
4. Lord Shiva as Narayana.:
तपसा प्राप्य सर्वज्ञं तुष्टाव पुरुषोत्तमः।।
महेश्वराय देवाय नमस्ते परमात्मने।। ७१.९६ ।।
नारायणाय शर्वाय ब्रह्मणे ब्रह्मरूपिणे।।
शाश्वताय ह्यनंताय अव्यक्ताय च ते नमः।। ७१.९७ ।।
“After practising penance he approached the consort of Umā along with Devas and eulogised Visnu said : Obeisance to you , to Lord Mahesvara , to the great Atman , obeisance to Nārāyana , to Sarva , to Brahma , to one having the form of Brahman ; obcisance to the permanent infinite one , to the unmanifest one .”
(Said by Lord Vishnu himself to Lord Shiva)
The name Narayana is applied to both Lord Shiva and Lord Brahma.
अहं नारायणो गौरी जगन्माता सनातनः ।१६.१५८
" I am Narayana , the eternal lord . I am Gauri , ' the mother of the universe ', I am the Oldest One“.
(Said by Lord Shiva Himself)
So, here again the name Narayana is applied to Lord Shiva and Maa Uma.
- In the Kurma Purana, Ishvara Gita, Lord Shiva says:
He who is called Narayana, the infinite one, the immutable source of all worlds and who is only another form of mine, performs the duty of protection.
(Chapter 15, 896-90a)
Those who follow Vedas look upon Lord Rudra and Lord Nārāyaṇa identical and they attain liberation .
(Said by Lord Brahma).
नामानि तव गोविन्द यानि लोके महान्ति च |
तान्येव मम नामानि नात्र कार्य विचरण ||
“Whatever names you have oh Govinda which are famous as Sahasranaamas , they are my names only without doubt”
These are the words of Maheshwara in Harivamsa. This means that all the names of Vishnu including Narayana is also Lord Shiva's name as said by Lord Krishna himself to Arjuna..
- Mahabharata Shanti Parva.:
“If that deity of deities, viz., Maheswara, be worshipped, then O Partha, is the puissant Narayana also worshipped. I am the Soul, O son of Pandu, of all the worlds, of all the universe. Rudra, again, is my Soul. It is for this that I always adore him. If I do not adore the auspicious and boon-giving Isana nobody would then adore my own self…”
“.He who knows Rudra knows myself, and he who knows myself knows Rudra. He who follows Rudra follows me, Rudra is Narayana. Both are one; and one is displayed in two different forms. Rudra and Narayana, forming one person, pervade all displayed things and cause them to act. No one else than Rudra is competent to grant me a boon. O son of Pandu. Having settled this in my mind, I adored in days of yore the ancient and puissant Rudra, for obtaining the boon of a son.”
(MBH 13:17:96).: praises Rudra as “ūrdhvaretā jale śayaḥ |” which means, “Thou ownest for thy offspring, beings higher than men and deities (viz., Brahma and Vishnu). Thou art stays in the form of that Vishnu who floats on the waters after the universal dissolution (as Narayana)”.
नारायणोऽहं नरकान्तकोऽहं पुरान्तकोऽहं पुरुषोऽहमीशः । अखण्डबोधोऽहमशेषसाक्षी निरीश्वरोऽहं निरहं च निर्ममः॥१७॥
17. I am Narayana, I am the destroyer of the (demon) Naraka, I am Ishana (Siva), the destroyer of the three (aerial) cities, I am the Purusha, I am the supreme Lord; I am the indivisible consciousness, the witness of all; I am without a superior, I am devoid of ‘I-ness’ (egotism) and ‘mine-ness’ (possessiveness).
- Rig Veda Khilla Khanda Śakala Śakha.:
ओं॒कारं च॑तुर्भु॒जं लो॒कना᳚थं नारायणम् ।
सर्व॑स्थि॒तं सर्व॑ग॒तं सर्व॑व्या॒प्तं तन्मे॒ मनः॑ शि॒वस᳚ङ्क॒ल्पम॑स्तु ॥२४॥
He (Lord Śiva) is Oṃkara; with four arms; patron of all the worlds; he is Nārāyana (indweller of the waters); he is stable and still in all, having entered all, pervading all; may my mind aspire towards Lord Śiva (or) Auspicious Śiva thoughts.
Conclusion.: With the above verses and the link contents, it is hense proved that Narayana is a common noun applicable to all the deities. Same as the names Purusha, Shri, Prajapati, etc etc.
I hope this clarifies all your queries. Prd..