好き is an adjectival noun. It is that, whether you see the な or not. The correct way to connect an adjectival noun to a noun it's modifying, is with a な. The correct way to connect other kinds of nouns is most often with a の (or else である).
You can say 猫が好きあの女の人は私の友達です, but if you do so you've inserted the additional word あの ("that woman"), which may not be appropriate in all the same places the original is, and has a slightly different meaning. This sentence is also technically grammatically wrong without the な, but I feel like it could be used this way in "real" Japanese. Reordering it to あの猫が好きな女の人は is more correct, though.
猫が好きな女 = Woman who likes cats (or possibly: whom cats like)
画家の女 = Woman who is a painter