Not to be pedantic, but he really isn't a carpetbagger. I mean, he's absolutely the type, but "carpetbagger" specifically refers to a politician who moves into a geographical area to run for office. The carpetbag was a type of cheap suitcase, meant to conjure the image of a hurried relocation for personal gain. In that sense, a carpetbagger is an opportunist and egocentric, which definitely describes Trump. And he did move from New York to Florida, but he never ran for local office as a Floridian. He was elected President, and is running for that office again. He's always been from the United States. He could just as easily run for president out of New York or New Jersey or New Mexico. His political identity isn't related to where he lives.
Yes, that's how it originated, and it was further used to describe any opportunistic politician moving to another geographic region to win elections there.