You only use "donner sa langue au chat" in the meaning of: giving up trying to guess something
So "cat got your tongue" would be as Jeanbar said: "tu as perdu ta langue?" or alternatively "tu as avalé ta langue?"
"cat got your tongue ?" = "lost your tongue ?" Meaning "tu as perdu ta langue "? (you ask someone a question and he/she doesn't answer, or for just asking why someone doesn't speak up) (cf Jeanbar, Maped , Wildan)
"donner sa langue au chat" means "give up" (cf: Wildan)