First, crâne de piaf literally means “sparrow skull”, not “skull sparrow”. But that would not be an accurate translation: piaf is a slang term for any kind of small bird. It often alludes to the fact that birds have small heads, and thus connotes a lack of intelligence. Crâne (skull) is simple metonymy for head and transitively for brain, and commonly used thus in slang expressions. So crâne de piaf means “small brain”, i.e. someone who is not intelligent. It's an insult. English has a similar expression in bird brain.
I would a character nicknamed Crâne de piaf to be especially stupid. Of course the expression could be used in an ironic way.
Linguee's database is built from “serious” texts (it's strong on legal and technical documents), so it isn't very useful for slang. Reverso indexes more fiction, and it finds many occurrences of crâne de piaf.