It certainly isn't difficult to obtain an upper bound of $n\lceil \log_2 n \rceil = O(\log_2 n!)$. All you need to do is to separately learn each bit of each value of the permutation.

As Michael says, the next natural question is whether you can obtain $(1+o(1))(\log_2 n!)$. I agree that it seems similar to the asymptotic question of sorting with comparisons. However, in the usual version of that problem, you are allowed to move elements based on an incomplete list of comparisons, which then amounts to adaptive comparisons. My guess is that it would be very cumbersome to attain a bound as good as $(1+o(1))(\log_2 n!)$ for this new question, which can be called "non-adaptive Mastermind for permutations with matrix guesses". Michael also suggested the restricted problem of permutation guesses, which to me seems even harder.

It certainly isn't difficult to obtain an upper bound of $n\lceil \log_2 n \rceil = O(\log_2 n!)$. All you need to do is to separately learn each bit of each value of the permutation. This looks like it is only roughly optimal, but actually it is already $(1+o(1))(\log_2 n!)$. (Thanks to t3suji for noticing this.) However, the $o(1)$ term only decays logarithmically in this algorithm, so you could ask whether you could make it $(1+O(n^{-\alpha}))(\log_2 n!)$.

I agree with Michael that it seems similar to the asymptotic question of sorting with comparisons. However, in the usual version of that problem, you are allowed to move elements based on an incomplete list of comparisons, which then amounts to adaptive comparisons. You could call this new question "non-adaptive Mastermind for permutations with matrix guesses". Michael also suggested the restricted problem of permutation guesses, which seems like a more difficult problem to me.