Fun fact: you can use this to play chess960 (i.e. fischer random chess), or any other "custom start position" variant, just by appending a FEN to the URL (replace spaces with underscores). Example: https://chess.maxmcd.com/bbnqrknr/pppppppp/8/8/8/8/PPPPPPPP/...
Chess960 also has different castling rules from normal chess though, and the site can't know it should use those just from the FEN?
That depends - Chess960 doesn't have different castling rules from normal chess, it just has additional considerations that normal chess does not.
In other words, Chess960's castling rules are completely consistent with normal chess castling rules, so depending on how it is implemented, it might just work.