He ran as one at the beginning, IIRC.
I find that you are right: after a lifetime as a Democrat, Bloomberg became a Republican hoping to siphon off some of the popularity that had accrued to Rudi Giuliani, but when convenient, he dropped his Republican affiliation like a Sophomore drops a Saturday morning class. He toyed with bringing his nanny-state loving, global-warming-smarmed personality to the Presidential race as a Republican, but that notion was drowned in bitter tears after the unwashed rejected the jackass out of hand.