I gather there’s a pun.
“Je donne ma langue au(x) chat(s)” as your dictionary says a colloquial phrase meaning “I give up guessing, just tell me”. But in the context you describe I expect it is not used in the figurative way but strictly means the girl has given her tongue to the cat, therefore she’s lost it and can’t talk.

“Donner sa langue au(x) chien(s)” has the same meaning and I don’t think it is really used nowadays.
See French dictionary here for meaning
and here for a possible origin.

« Jeter sa langue au chien » est une très ancienne expression puis elle a évolué et s’est transformée en « Donner sa langue au chat » pour que ce soit moins barbare.