i thought pin was the nickname the nurse gave it because of its little erection the nurse used to fap herself

To add to what everyone above has said, it's actually an ironic name, as instead of Pin(nochio) becoming a "real boy" in the end, the "real boy" becomes the dummy.

Pinocchio (b/c he is a dummy)

And maybe given the fact that Pinnochio never lied, so Pin is short for Pinnochio like his nose never grows longerPin-nochio

Ursula's boyfriend said that she told him she named him Pinnocjio at age 3.I'm guessin Pin is short for Pinnochio