The "Tathacharya" referred to in the Wikipedia link is a well known acharya in the Srivaishnava Sampradaya. I am not aware of an explicit mention of Krishnadevaraya being initiated into Srivaishnavism but both Ahobila and Parakala Mutts are known to have had a long association with the Vijayanagar empire in general and Krishnadevaraya in particular (but that by itself is insufficient evidence). If I find a source, I will update this answer.
That said, that Krishnadevaraya's personal affiliation was towards Srivaishnavism is unambiguously evident from his work "Amuktamalyada" - a telugu rendering of the story of Andal who is one of the 12 Azhvars.
I will cite two scholarly works that establish this:
According to Velcheru Narayana Rao et.al., among the characters referenced in the Amuktamalyada, other than Andal, we find references to Periyazhvar (aka Vishnuchitta) and Yamunacharya (pre-Ramanujan acharya of the Srivaishnava Sampradaya). More interestingly, the authors note that Krishnadevaraya incorporated Srivaishnava values into governing principles.
in tone. A distinction is drawn -- perhaps for the first time in South
India -- between the king as individual, with his individual
inclinations and exigencies, and kingship as institution (which has to
go on at all costs). A basis for stable kingship is elaborated around
Srivaishnava, trans-local values, with a yogic colouring, an aesthetic
component (expressed through music), and strong themes of personal,
Krishnadevaraya eschews the standard solution, namely the resort to
purāṇic and dharmaśāstric normative language regarding an alliance
between Brahmins and Kshatriyas. Instead, his preference is markedly
for yogic and renunciatory themes that are at the same time strongly
and paradoxically allied to a Srivaishnava idiom rooted in the idea of
bhoga (enjoyment) – a theme to which we shall return below
and in the conclusion
It may be argued that the rājanīti section of the Āmukta-mālyada
embodies a paradox, since it gives pragmatic advice on rulership while
at the same time being framed in a text where the ideal that is
insisted upon is one of renunciation. The paradox is however easily
resolved. For the central thrust of the Srivaishnava political
theology in which this advice comes embedded is that it is necessary
to separate rulership as a social function – which is not merely
desirable but essential for the world to cohere -- from the personal
salvation of the ruler. It is understood that at some moment or other
in his life, the king will wish to escape the bonds of his royal life
and become a renouncer. The text does not suggest otherwise, and quite
on the contrary, expresses its unbounded admiration for such rulers.
But the view is clearly that renunciation is not a solution to the
problems of kingship, only to the problems of the king as individual.
We are not quite in the realm of the ‘two bodies’ of the king
developed by Kantorowicz, but what is obviously being proposed here is
a view where Srivaishnava this-worldliness plays a central role in
protecting the king from renunciatory excess.
In his PhD dissertation submitted at UC Berkeley, "The Āmuktamālyada of Kṛṣṇadevarāya, Language, Power & Devotion in Sixteenth Century South India", Srinivas G. Reddy says:
Kṛṣṇadevarāya was thoroughly steeped in the Śrīvaiṣṇava tradition and
almost every poem in his great work exudes a sense of his personal
devotion. The famous avatārika padyam or invocation verse to his
iṣṭa-devata Lord Vĕṅkateśvara is also interpreted as a salutation to
Kulaśekhara Āḻvār in his aṁśa-avatāra as the Kaustubha gem. The
invocatory decad that ends with an explicit poem about the saints
(quoted above), is in fact a sustained evocation of their divine
status, and their central importance in the poet‟s heart.