Good question, but no.

A cookie is one of the types of biscuits.

Cookie monster, to my knowledge, normally ate cookies in the English sense, so the name still worked either side of "the pond".

wiki article - chocolate chip cookie

We probably just accepted that he occasionally ate other biscuits too - but also, we grew up with Standard Definition TV, so maybe there just weren't enough pixels to tell what he was eating.

We call it greedy bastard

It?? It?? How dare you call our lord and saviour it haha