Calling a person Babylonian is rather like calling someone a US citizen or a British citizen; it refers to their nationality rather than their religion. This is of course of immense importance when one is applying for passports or for support from the state administration. Of course, there was no such thing as nation states then, but something similar will have applied.
Abraham, like his father, was born in the city of Ur of the Chaldees, according to the Biblical narrative; and in 1927, Leonard Woolley identified this city with the Sumerian city-state of Ur founded in 3800 BCE and located now at Tell el-Muqayyar in southern Iraq's Dhi Qar Governate. At the time of its founding it was a coastal city near the mouth of the Euphrates but this has now shifted and do the city is now well inland, being around 16km from the river.
It's probably inappropriate to call Abraham a 'citizen', as this again is a modern term; Ur, although a city state, was a monarchy, with the first king of the first dynasty called Mesannapada; and the city's patron deity is Nanna (or Sin in Akkadian), the Sumeruan moon-god and from whom the city takes its name: Unug, literally, the abode of Nanna.
Whilst being a 'citizen' of Ur was important in terms of where he lived and how; this is, however, in the temporal order of things and the Bible is less concerned with this order than the spiritual order.
It's in this sense that we think of Abraham as primarily being the patriarch of Judiasm, Christianity and Islam; rather than Sumerian. What we can say is that the biblical Abraham, like his father Terah, was born in the Sumerian city-state Ur, in modern day Iraq, and was told by God to emigrate to the land of Canaan which had been given originally to Canaan, a son of Ham, and a grandson of Noah; and was now promised to him; and according to the Table of Nations in Genesis 10: it comprises all the land from Sidon or Hamath to the north of Gaza and Lasha to the southeast; in modern day terms, it is known as the Levant, and comprises all of Israel, Palestine, Lebanon, Western Jordan and Western Syria.