Cross-breeding...?

[Belhi]

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.

Dragons are Asexual. They don't pick mates for breeding cause they don't need mates to breed cause they don't need a mate to breed. Dragons don't use breeding to create diversity but rather go through substancial physical evolution during their lives.