An exact formula in which the largest term gives an asymptotic formula is formula (5) of mat.univie.ac.at/~kratt/artikel/encystat.pdf.

See, e.g., math.stackexchange.com/questions/36798/… and stackoverflow.com/questions/6240113/….

The problem of counting these matrices up to row and column permutations is solved in I. M. Gessel and J. Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591-612, available online at arxiv.org/abs/0705.0042. (In the paper they are called "semi-point-determining bicolored graphs".) Of course, as in most unlabeled graphical enumeration problems, the formula is not simple, and might not be so helpful in deriving an asymptotic formula. The numbers in Richard's formula are sequence A181230 in the OEIS.

You're right. I rewrote it to use the q-binomial theorem.