TTS Valencia talks about the destruction of the Umayyads' history after their arrival. My guess is that they had some superficial things in common with the ancient group(enough for the Castilians to apply the name), and after a few generations of intermarriage with conquered Castilians, Castilian stereotypes were all they had left.
It's possible they didn't even have a name for themselves beyond "Us". By all accounts they were originally a very tiny group, and how many of us folks actually bother to come up with a name for their closest circle of friends and family?