The game is officially, "soccer" in the U.S. [United Sates Soccer Federation] and Canada [Canadian Soccer Association].
FIFA calls the game beach soccer, "Beach Soccer." It is played under the "Beach Soccer Laws of the Game."
This definition applies to the FA per it's constitution and bylaws.
"All Clubs and Aiffliated Associations shall play and/or administer football in conformity with
these Rules and also:
(a) The Laws of the Game (as deﬁfned in the Articles); and
(b) the statutes and regulations of FIFA and UEFA which are in force from time to time."
[THE RULES OF THE FOOTBALL ASSOCIATION LIMITED, page 1]
Here is a list of references to "association football" on the FA's Website: https://www.google.com/search?q=site%3Athefa.com+%22associat...
"Beach Soccer" is of course relevant because that's the name of the game sanctioned by FIFA.