My question is motivated by the recent question and more recent appearance of its author Bruce Westbury. Most of you know that the best way to find a sequence of integers is looking for it on The On-Line Encyclopedia of Integer Sequences. (My personal success of using this powerful database is however less than 10%.) Another strategy (especially, when you suspect that your sequence is holonomic) would be trying a guessing package in a computer algebra system, like $\operatorname{gfun}$ in Maple.

*What can we do if we need to identify a sequence of polynomials?*
(for simplicity with integer coefficients)

One recipe (which was used by my colleague for solving the problem in this question) is again to use the OEIS, since the latter contains many 2D examples as well (like the whole Pascal triangle of binomial coefficients and several subcollections from it). The chances are miserable (as Bruce's sequence shows). Even having some additional information (like knowing that the polynomials are $q$-analogues of a known integer sequence), there seems to be no general machinery or database to assist in identification. Are there algorithms (better implemented) for polynomials analogous to $\operatorname{gfun}$?

Thanks!

*P.S.* The tag "soft-question" here means that the question is indeed soft
but also on software.