Game Michael is pretty much always seen self identifying as Mike, like on the keypad in Sister Location, writing his name in the Survival Logbook and Ticket to Fun, and using the name Mike Schmidt in FNAF 1. I always believed this was because Michael was the name William gave him and the one William used the most on him. He started going by Mike to try to put some space there and gain some independence, and as an act of defiance and reclaiming some of himself. This makes sense seeing that the only time we see him self identify as Michael is when talking to William. And I know many other people believe this as well. Which is why it was so jarring to see in the FNAF 2 movie when the investigators show up and address him as Mike, and he goes out of his way to ask them to call him Michael, because in this universe he is proud of the name his benevolent father gave him. It's actually kind of maddening when I think about it for too long because movie Michael is like everything game Michael would hate to have seem himself become.