Although not a literal translation , "Would you like" is used in English interchangeably with "Do you want". Shouldn't that also be a correct response?
If you wanted to ask "would you" you would need to use "vorresti / vorrebbe" (2nd/3rd person respectively).
I agree it is more commonly used in English in general.
...we know. It's not changing gender dude, it's being conjugated.
In PliLopes's example, he isn't using a plural of "giocare", he is doing the "tu" form of "to play" (you play)