On the same lines, are there any claims about other puranas claiming themselves or others as the best, foremost etc.?
Like Bhagavatams, Shiva Purana also says itself as best among all the Puranas and best remedy in age of Kali. Chapter 1 Glory of Shiva Purana says
कालव्यालमहात्रासविध्वंसकरमुत्तमम्। शैवं पुराणं परमं शिवेनोक्तं पुरा मुने॥८॥
This Purana is the best remedy for the biggest danger like the death, which bites like the snake and is the biggest danger. The Šiva Purana is the best of all the Puranas. O Great ascetic, initially it was spoken by Lord Siva himself.
एतस्मादपरं किञ्चित्पुराणाच्छैवतो मुने! विद्यते न मनःशुद्धयै कलिजानां विशेषतः॥ १०॥
Except the Siva Purana, there is no other remedy in the age of Kali, to purify the minds of the people.
जन्मान्तरे भवेत्पुण्यं महद्यस्य सुधीमतः। तस्य प्रीतिर्भवेत्तत्र महाभाग्यवतो मुने!।॥ ११॥
O Best of the sages, only such of the intelligent person gets an opportunity to listen or read it, who has to his credit the noble deeds of the earlier births.
एतच्छिवपुराणं हि परमं शास्त्रमुत्तमम्। शिवरूपं क्षितौ ज्ञेयं सेवनीयं च सर्वथा॥ १२॥
The Siva Purana is admittedly the best of the Sastras and is also the true form of Siva on earth, therefore, it should be read, narrated, listened studied and well thought over.
Yes , in Shiva-Mahapurana we also find the similar description in Mahatmyam or Glory part. Chapter 1 Shloka 12 - The glory of the purana.
एतच्छीपुराणं हि परमं शास्त्रमुत्तमम्।
शिवरुपं क्षितौ ज्ञेयं सेवनीयं च सर्वथा॥१२॥
The Shiva-Purana is admittedly the best of the Shastras and is also true form of Shiva on earth.Therefor it should be read narrated , listened , studies ans well thought over.
Similarly, Vishnu Purana also states Matsya Purana as the greatest:
मुख्यं पुराणेषु यथैव मात्स्यं स्वायंभुवोक्तिव्स्तपी संहितासु।
मनु:स्मृतिनां प्रवरो यथैव तिथीषु दर्शो विबुधेषु वासव:।। (Vishnu Purana Chapter 13 verse 48)
As Matsya purana among puranas, Code laid down by Swayambhu among the codes, Manu Smriti among Smritis, Amavasya among Tithis and Donation made on Visuvas (rotation of Aries and Libra) among other donations is greatest.
In the above shloka Vishnu purana is declearing Matsya purana is greatest among puranas.
Just like Bhagavatams and Shiva Purana, even Agni Purana considers itself as the greatest of all Puranas.
Chapter 383 - The Glorification of the Agnipurana, verses 44b-51 say:
44b-47a. I have heard the Agneyapurana with respect by the favour of Vyasa. The Agni (purana) is a manifestation of the Supreme Brahman. You and other sages such as Saunaka and others had worshipped lord Hari in the Naimisa forest. They were steadfast. The Agnipurana is spoken by Agni and conforms to the Vedas. This (Purana) endowed with the two (sorts of) brahmavidya confers enjoyment and emancipation.
47b-51. There is nothing richer in substance than this. Nothing is a better friend than this. There is no greater work than this. There is no better course than this. There is no greater treatise than this. There is no greater scripture than this. There is no superior knowledge than this. There is no smrti (textbook of piety) than this. No agama (texts dealing with worship) is superior to this. No knowledge is greater than this. No exposition excels this. There is nothing extremely auspicious excelling this. No philosophic exposition is greater than this. This is the foremost Purana. There is no object more diffcult to obtain than this on the earth. All the lores have been expounded in this Purana.
Kalika Purana is the supreme Puraan and real Bhaagvat Puraan in Tantrik tradition. In one translation on archive (https://archive.org/details/KalikaPuranamVol.1AcharyaMrityunjayTripathi/page/n31/mode/2up) you can see Markandeya says:
आप सभी मुनिगण मेरे द्वारा कथित अत्यंत गोपनीय से भी गोपनीय, पवित्र, शुभकर्ता, सम्यक, ज्ञान प्रदान करने तथा कामना की पूर्ति करने वाले, श्रेष्ठ, इस कलिका पुराण को सभी सुने Which basically translates to -- All you munis, listen to this extremely hidden, sacred, auspicious, complete/correct, knowledge giving and wish fulfilling, supreme, Kalika Purana.
Here is another abridged version of the Purana in Hindi - https://archive.org/details/HindiBookKalikaPuran/page/n3/mode/2up . It also says that this scripture is supreme:
यह लिखा हुआ उत्तम पुराण ... यह परम तंत्र है ... इससे अधिक अन्य कुछ भी नहीं है
Which means that this is the best purana ... the supreme tantra ... there is nothing more other than this (Kalika Purana).
People like Hemadri in Chaturvarga Chintamani (13th century) also used to call it the real Bhagavata. There is also another verse in Skanda Purana, Kedara Khanda 1. which hints on this subject:
अष्टादशपुराणेषु दशभिर्गीयते शिवः। चतुर्भिर्भगवान् ब्रह्मा द्वाभ्यां देवी तथा हरिः॥ ‘अठारह पुराणों में दस में शिव-स्तुति है, चार में ब्रह्मा की और दो में देवी तथा हरि की है।’
Which means -- Out of the 18 Mahapuranas there are 10 of Shiva, 4 of Brahma, 2 of Devi and Vishnu each.
So the interpretation is
- Kalika Purana is called the supreme scripture
- It is focused on Bhagavati
- It could be a Mahapurana mentioned in Skanda Purana of Shivite sect
The Brahma-Vaivarta Purana Also declares itself as the best of all puranas.
Brahma Vaivarta Purana, Krishna Janma Khanda, Chapter 113:-
The Varahapurana Also declares itself to be the greatest of all sastras.
Varaha Purana Chapter 148
Narada Purana also declares itself as the best of all Puranas.
Narada Purana IV.125.47
At another instance it also declares the Lingapurana to be 'greater than all other puranas'.
Narada Purana IV.102.4
Like All the other examples provided, Brahma Purana too calls itself better than all other shAstrAs
पुराणं वैष्णवं त्वेतत्सर्वकिल्विषनाशनम्। विशिष्टं सर्वशास्त्रेभ्यः पुरुषार्थोपपादकम्।।
This is a Purana belonging to Visnu. It destroys sins. It is more excellent than all other scriptural texts. It is conducive to the fulfilment of all aims of life.
~ Brahma PurAna 138.20
In Shreemad Bhagavatam 7th Chapter of 12th Canto. Each and Every Mahapurana is best to read because Adishakti, Brahma Shiv, Vishnu gives good wishes for us next Stage of Life.
In Shri Vishnu Purana 6th Book 2nd Chapter of 15 - 16 Shloka
In Garuda Purana 227th Chapter of Karma Kand 9 - 11 Shloka respectively VyasaJi states Kaliyuga is Yuga of evils ( Human ). Kali Yuga is best because In Satya 10 years of Tapasya, Treta 1 year of Tapasya, Dwapar 1 month of Tapasya, Kali 1day of Tapasya they will get there fruit.Those persons are good and luckiest ( Because they do More than thousands of year of Tapasya ) which (reads/listens/remember/re-sign/spend Everything/Kirtan/cries/Bhakti) name of Supreme One. Because in other Yuga they have weight and pain of many things. In Kali Yuga, they will weightless and painless from attraction of materialism.