Yes, crossbreeding can--and often does--happen. It may be socially frowned upon, but there is plenty of evidence that it happens. Here be Heavensward Spoilers:

One of the biggest pieces is the Mongrel, who is the illegitimate offspring of a highborne Ishgardian Elezen and one of his Hyur vassals. We also have Shiva, who fell in love with Hraesvalgr.


I suspect that the reason we don't see more of it is the fact that it would require more character model customization options than anyone has time to create or code.