In ancient Mesopotamian cultures (as in most ancient cultures) religion wasn't separate from daily life at all. (Such separation is a relatively new development in human history.) Since the religion was not separated, it didn't have a name (that we know of.) Names are words we use to differentiate, and the religious life of a Mesopotamian person was non-separate.

As far as cultural identities- there were varying levels of these. Mesopotamia is a pretty large area, and contained many different cultures, some overlapping. The cultures often had different names for one another- for instance, the word "Sumer" which we use to describe a certain culture, is not a Sumerian word at all; it's an Akkadian word (Shumer) meaning "black-heads." Sumerians called their land "Kiengir". ("Kiengir" can be translated a couple pf ways- it could be "Land of the Foremost of the Gods" [ki=land, en=lord, gir=gods] but it could also be "Land of the Lord of [various words associated with conflict and opposition.]" The former is probably the correct translation, but I find the multiple meanings in Sumerian language to be fascinating and often clues to their culture, as puns abounded and many important mythological stories are imbued with double entendre.) We do not know what the Sumerians called themselves, although it was probably something equivalent to "the people."

It is possible that Mesopotamians identified themselves in some way with the personal god whom they worshipped. Each individual in Sumerian and Babylonian culture had a "personal god"; for some this was one of the conventional gods, those part of the pantheon, while for others it seems to have been what Kramer called a "good angel" (and what modern Hermeticism calls the "Holy Guardian Angel." However, we don't have enough information to know if there were specific names for worshippers of certain gods (although it's likely, because Sumerians had names for almost every strata of society.)

National groups- as I mentioned, Mesopotamia was a very divided area. Not only were there different ethnic groups, and different, overlapping cultures, but different city-states also, which were sometimes united, but sometimes at war with one another. So, there were no national groups per se, but there were many of what we would today consider nationalities. There is a LOT of information about this, albeit slanted toward the Akkadian/Babylonian historical angle, on the Gateways to Babylon website.

I hope that this helped to answer some of your questions!