How many chess variants are recognised?

The possibilities are endless! The site linked below lists 119 variants, including the base game.

https://greenchess.net/variants.php?list

According to FIDE's Laws of Chess, there are just two variants: Standard Chess and Chess 960, the latter being a variant where the starting positions are, within certain limits, randomized to make the game more about on the spot tactical thinking and less about memorizing prepared and proven opening lines.

Both variants are further subdivided by time controls - Classical, Rapid and Blitz - but those do not change the game as such, only the way certain tournament rules are enforced.

https://fide.com/FIDE/handbook/LawsOfChess.pdf

All other variants are casual and not "recognized" by an official chess authority.

There is no single, authoritative number of officially "recognised" chess variants because they are often defined, promoted and played in different communities and not governed by a unified global body.

David Pritchard's "Encyclopedia of Chess Variants" (1994) documented over 1,400 variants. The more recent and expanded "Classified Encyclopedia of Chess Variants" (2022, by Pritchard and John Beasley) describes more than 2,000 variants.

https://en.wikipedia.org/wiki/David_Pritchard_(chess_player)

Online databases like chessvariants.com list well over 2,000 distinct variants, including user-submitted games.

https://www.chessvariants.com