Though there are many deities, these can be generally considered as falling under the broad classification of Saiva and Vaishnava. Between Siva and Vishnu, he may have thought about which to choose. The Acharya was himself an Avatara of Siva. So, he may have thought-why use my own Name. Siva is very dear to Vishnu. And, similarly, Vishnu is very dear to Siva. Therefore, wherever Vedanta refers to Paramatma as Jagat-karana-sakthi, Prapancha-mahasakthi, Saguna-bhramman, Iswaran, with an implicit suggestion of attributed Form, where Murthy Rupa is indicated, the Acharya has used NARAYANA as the name..
In between, I mentioned the word Iswara. Though Iswara is commonly equated with Siva, in the Vedanta Sastra, Iswara is Sagnuna-brahman. Ramanujacharya and Madhwacharya also have used the word Iswara in this sense only. The Acharya has composed many stotras on Siva, like Sivananda Lahiri. It may be asked whether in those stotras he was doing stuthi of himself. They are bhakthi stuthis. Different sects and communities of people have their own likes and preferences among the deities. In order to be helpful to one and all in concentrating and steadying their mind in the worship of their preferred God, the Acharya composed stuthis on all the deities. How could he have left the important deity Siva? So, he composed stuthis on Siva also.
The Acharya had no "identity" for himself. He was always "Atma Swarupa". Therefore, it was possible for him to take on, as it were, identities with different mental inclinations and work out appropriate ways for the elevation of people with different mental inclinations and preferences. Water, which is colourless, appears black when passing over black soil, and red when passing over red soil. It is like that. So, for the benefit of people who take to Bhakti marga for self-fulfillment, he made stuthis to different deities to suit different preferences. When he made such stotras, he became, as it were, the Bhakta of such deities. In that way, when he made stotras on Siva, he could become a Siva Bhakta and not remain a Siva Avatara. It would be self-praise only if, remaining a Siva Avatara, he sang in praise of Siva. If he became a Bhatka of Siva and sang his praise, did it not become permissible to him to sing his praise as much as he wanted ?
It may occur to some that the Acharya gave central place to Siva, as Chandramouleeswara, in his Mutts and made the other deities adjuncts. The Mutts are meant for all and so it may be asked why he gave central place to Siva in the Mutts. The answer is that the Acharya did not plan that this should be so. The Parameswara (God) himself gave to his Avatara five Spatika Lingas and commanded him to have them installed in five places and arrange for the pooja. In carrying out this command, the Acharya installed in both his Mutts, here in Kanchi and in Sringeri, two of the Spatika Lingas. Thus the Lingas that Parameswara himself had given and installed in these two places for which poojas were arranged became the central deities of worship in these Mutts and other deities became parivara deities. In the remaining Mutts also, for the sake "uniformity", the Acharya had necessarily to make the pooja arrangement as Siva Panchayatana.
Now you may ask: You said that Acharya, when he wrote Bhashyas which were books on Atma Sastra, was only in "Atma Swarupa" and had no "identity"; if he was "Atma Swarupa" , and remained as such, then why should he identify with Siva Avatara, and therefore think about "not talking about himself" and therefore think about using the NARAYANA name. I knew such a question can arise. I have my answer ready. I came to talk about this with a question paper and a answer sheet, all prepared.
Continued in Namo Namah part 3 of 4