Your intuition is correct: a recursive loop with the state as a parameter is the standard Erlang approach.
This concept is often abstracted away by using one of the server behaviors in OTP.
Simple example, may contain errors:
game_loop(X, Y) ->
receive
{moveto, {NewX, NewY}} ->
notifyClient(NewX, NewY),
game_loop(NewX, NewY)
end.