USCF has stipulations allowing for the game to continue after an illegal move has been made. FIDE tries to avoid such situations.
But it's entirely possible for an OTB game to be played from an illegal position. Heck, Chess960 is all about starting play from otherwise illegal positions.
In an illegal position (opponent leaves king hanging), you could make a move which would result in a mate, while not exposing your king nor moving the piece in forbidden way. That creates a mate position (opponent cannot escape check legally), therefore ending the game and your win gets written on the scoresheet, as a situation which ends the game has occured.
Illegal moves do not end game on their own, mate does.
But those positions are not illegal in Chess960...
It's funny, I was about to make an argument that Chess960 allows for different castling moves... but the game does not care about where your king and rook start, if you never move them and then place on g1+f1 / c1+d1, it is always the same castling.
Related question: what if a legal position is reached via illegal moves? Do legal moves exist in that case?