As far as I know LaVey was a Jew whereas he claimed to be a Gypsy or whatever.
IMO his "religion" was a synthesis of Ayn Rand and Ragnar Redbeard (and thought Redbeard was a pseudonym of Jack London, since disproven I think?). And just a touch of Crowley.
The "Satan" is mostly just a symbol. There was no literal Satan, although he claimed some proficiency in magic(k) too.
Everything about him and his supposed life story was fake but this could be called a form of Satanism or Occultism too since what matters is manipulation and control is what matters, "if it works," not so much the exact mechanisms behind it.