It is complicated. Tom Waits won a case where a commercial imitated his style.
Not under US copyright law, AFAICT. He's won cases where he sued over things like false endorsement - i.e. claiming that listeners would believe it was actually him, not just that the style was similar.
(Apparently he did win a similar case in Spain under their copyright law, but from skimming articles it sounds like the issue there also was impersonation, not just stylistic similarity.)