Is this a reference to a TV show, or to a playground game?
TV show. Based on "The Farmer Wants a Wife".
Bauer sucht Frau?
Ja.
Is it a Polish show?
"Rolnik szuka żony" is a Polish TV show, although the original "Farmer Wants a Wife" is from the UK.
Never heard of either. But now I have 'The Farmer in the Dell' rolling thru my head lol
That's a neat reference right here!
Szuka takes genitif?
Yes, it's one of those exceptions that you need to remember.