Squid is not classified as being a fish because of the fact that it is actually a cephalopod. Fish however are defined as water-dwelling vertebras which have gills which squid do not. There are quite a few differences between fish and squids, both in terms of how they are classified and physical characteristics as well.
Squid is from the order of Teuthida which is part of many different species, around 300. Although many people lump sea-dwelling organisms like fish and squid into the same category, the fact of the matter is that they are just not the same. There are many distinct differences that separate these organisms from one another.