Is there a way I can claim it?

You have to wait for the account to clear, if it does clear. Usually this takes a year later after the frozen date, which you can view on the frozen users' profile. Exceptions to this include a user coming unfrozen, usually if someone was frozen wrongfully or they self-froze and decided to come back.

If the account does clear, however, you will possibly have competition with other users for said pet name though.

I suspected as much - sadly the user account the pet belongs to won't clean until next year, so I feel I might have to look for a more unique name for my pet (who ironically has his name stolen when I was inactive) and maybe exchange them closer to the time!