Even though a definite proof -- in the form of a confirmation by the Nolans or something similar -- is still missing, I think there is at least enough evidence to conclude that this connection is not coincidental, even if that evidence is only circumstantial.
First of all, you are definitely not the only one drawing that connection. There are various articles around the web suggesting that exact same visual inspiration for the Tesseract. Just a few examples are:
Furthermore, Borges in general, whose recuring themes among others often include questions about infinity and time, has been repeatedly identified, and confirmed by Nolan, as one of his many recurring influences (especially attributed to Inception), as also in an interview on Interstellar in particular:
...What fascinates you about interdimensionality?
It might be unusual in movies, but it’s very well established in other media. I’m very inspired by the prints of M.C. Escher and the interesting connection-point or blurring of boundaries between art and science, and art and mathematics. I’m thinking of his Penrose steps illustrations that inspired Inception. Also, the writing of Jorge Borges, the great Argentinian writer, wrote all kinds of incredible short stories that dealt with paradox. But I feel like films are uniquely suited towards addressing paradox, recursiveness, and worlds-within-worlds.
So Borges is definitely no stranger to Nolan. But as this interesting Wired article shows he goes even further as to include a copy of Borges' Labyrinths (a collection containing, among others, The Library of Babel) in Murph's bookshelf itself!
And last but not least, it also makes sense even apart from the obvious visual similarities of vastly infinite rooms of bookshelves. We just have to think about the library's (an obvious analogue to our own universe) vast and apparently infinite size which puts all the possible works ever written right at our hands, yet with no chance to take any actually useful knowledge out of this infinite mess of multisensical but ultimately nonsensical information without knowing where to look for. Now set this in connection to Cooper whose personal relation to his daughter gives him the means to understand this mess. He is the necessary index that enables the apparently omnipotent but ultimately insubstantial "Bulk Beings" to make sense out of the infinity of spacetime in the first place. Cooper basically is the "Man of the Book", as also reflected in his dialogue inside the Tesseract:
They have access to infinite time and infinite space, but they're not bound by anything. They can't find a specific place in time, they can't communicate. That's why I'm here, I'm going to find a way to tell Murph, just like I found this moment.
So even if an obvious confirmation by Nolan is still missing, and the Tesseract might not have been conciously planned as a deliberate homage to Borges' Library of Babel right from the beginning, I cannot imagine that this connection completely eluded such a Borges-enthusiast as Nolan. I would thus very much understand the Tesseract's visual realization and it's conceptual interpretation as a citation and homage to The Library of Babel.