Trump isn't considered a Republican by any of the powerful people in the Republican Party, or most of its big traditional financial backers (the Kochs particularly dislike him). It's why people like Mitch McConnell turned on him when it was clear Trump's power was gone. It's why they did everything they could to prevent him from winning the party nomination in the first place. The other Republican candidates were very weak and Trump sensed that and proceeded to smash them one by one.
The comedy of it all, is that Trump running was merely an arrangement with his former pals the Clintons (arranged in a phone conversation with Bill Clinton prior to declaring [1]). He was supposed to just target and knock off Jeb Bush as a favor. The reality TV style name attack gimmick (low energy Jeb, sleepy Joe, etc) worked so well that it became clear there was a runway to the Presidency and he took it.
Trump is a populist opportunist narcissist, all in one. Party is irrelevant to him, he isn't a partisan at all. He would have happily run as a Democrat - which is what he was previously - if he could have gotten to the Presidency easier that way.
[1] https://web.archive.org/web/20150819083634/https://www.washi...