Yāskācārya, the ancient author of the Niruktam, explains this mantra as below (7.18):
इममेवाग्निं महान्तमात्मानमेकमात्मानं बहुधा मेधाविनो वदन्ति इन्द्रं मित्रं वरुणमग्निं दिव्यं च गरुत्मन्तम् । दिव्यो दिविजो गरुत्मान् गरणवान् गुर्वात्मा महात्मेति वा ।
This same Agni, who is the one great Ātman, is described by the wise people in many ways, as Indra, Mitra, Varuṇa and the divine Garutmān. 'Divine' means 'born in heaven', and 'Garutmān' means either "one who swallows greatly" or "magnanimous or great Ātman."
So Yāska attributes the mantra to Agni as the great Ātman, who is manifest in different forms.
Sāyaṇācārya gives pretty much the same explanation, interpreting the Ātman as either Agni or Sūrya. He just goes into detail about each deity:
अमुमादित्यं ऐश्वर्यविशिष्टं ’इन्द्रं’ ’आहुः’ । तथा ’मित्रं’ प्रमीतेर्मरणात् त्रातारमहरभिमानिनमेन्नामकं देवं तमाहुः । ’वरुणं’ पापस्य निवारकं रात्र्यभिमानिनं देवमाहुः । ... कथमेकस्य नानात्वमित्युच्यते । अमुमेवादित्यं ’एकं’ एव वस्तुतः सन्तं ’विप्राः’ मेधाविनो देवतातत्त्वविदः ’बहुधा’ ’वदन्ति’ । तत्तत्कार्यकारणेन इन्द्राद्यात्मानं वदन्ति । "एकैव महानात्मा देवता स सूर्य इत्याचक्षते" इत्युक्तत्वात् । किं च तमेव वृष्ट्यादिकारणं वैद्युताग्निं ’यमं’ नियन्तारं ’मातरिश्वानं’ अन्तरिक्षे श्वसन्तं वायुं ’आहुः’ । सूर्यस्य ब्रह्मणोऽनन्यत्वेन सार्वात्म्यमुक्तं भवति । अत्र ये केचित् ’अग्निः सर्वाः देवताः’(ऐब्रा२।३) इत्यादिश्रुतितोऽयमेवाग्निरुत्तरे अपि ज्योतिषी इति मत्वा अग्नेरेव सार्वात्म्यप्रतिपादकोऽयं मन्त्र इति वदन्ति । तत्पक्षे प्रथमोऽग्निशब्दः उद्देश्यः । तमग्निमुद्दिश्य इन्द्राद्यात्मकत्वकथनम् । अयं मन्त्रः निरुक्ते एवं व्याख्यातः - "...." इति ।
This Āditya is called "Indra" when described as mighty. Then he is also called "Mitra" i.e. "one who protects from death" and the presiding deity of the day. Then he is also called "Varuna" i.e. "absolver of sin" and the presiding deity of the night.... How the one is called many is explained. This Āditya, being one, is described in many ways by the wise people, i.e. those who know the deity. They call him variously on account of the variety of functions. It is said "There is only one great Ātman, and he is called Sūrya". Moreover, he is called Lightning Agni as the cause of rain, Yama the controller, Mātariśvā the wind. Due to Sūrya's identity with Brahman, he is described as the Ātman of all. Here there is another view that Agni is all the deities (as said in Ait. Br. 2.3), and hence this mantra is explained on behalf of Agni (instead of Sūrya). Hence Agni is said to be Brahman and hence is described as the Ātman of all. Then, Agni is described as being Indra, etc. The Niruktam explains this mantra in this way."
So this much directly answers your question about traditional commentaries.
To your other statement about "The above verse has been popularly quoted by many Neo-Vedantins as to mean that all religions are true.", here is my meta-analysis:
The vast majority of Hindu scriptures were composed or canonized much much earlier than the advent of foreign religious systems such as Islam and Christianity. Definitely, the Vedas were finalized and closed by 2000 BCE [1], which is still very ancient compared to the time period when Christianity and Islam became visible in India. So the above mantra pertains to the Vedic pantheon of deities, and any deities derived from the Vedas and other allied scriptures (Puranas, Agamas, etc). The implication is that within the framework where different deities are viewed and mutually accepted as ultimately forms of the same Brahman, this philosophy of all gods being the same has validity. However, Islam and Christianity are outside the above framework, because they do not accept conceptions of God other than their own as equivalent to their God (i.e. "laa ilaah ill'allaah" and "No one comes to the Father except through me" https://www.openbible.info/topics/jesus_being_the_only_way, etc.).
Without using specific labels such as "Neo-Vedantins", we see vast sections of modern Hindus being ignorant of the above subtle differences because they have not studied other religions. This is because they have not updated their awareness to reflect new developments. Old scriptures will not have answers to such new developments. Ancient scriptures are great for rediscovering the unchanging eternal truths, but not for new challenges from new developments. Hence every new generation of Hindus should analyze the new developments and come up with new answers which combine the ancient wisdom with modern pragmatism. This is what every generation of Hindus have done when new challenges came up. Rajiv Malhotra has elaborated similar themes of "mutual respect" in his book "Indra's Net".
Footnote:
[1] Based on new research using astronomical references in Vedic literature. See:
http://koenraadelst.blogspot.com/2019/05/astronomical-chronology-of-vedic.html
https://www.youtube.com/watch?v=jfW4iLBgxg8
https://www.youtube.com/watch?v=8GA29oqlCko
https://www.youtube.com/watch?v=QAoVx2BPy9Q
