According to Rabbi Munk, Beor, the father of Bil'am, was the son of Lavan. There are meforshim who say that the wall that his leg was squeezed against was the rock that Lavan and Yaakov put up as "Gal Aid" to guarantee that members of either family would not cross to harm the other. According to this, he was of the same nationality as Lavan.
Rabbi Sacks (former Chief Rabbi of Britain) derives the name Bil'am from "Bli Am" (without a nation) and says that as a "Curser for Hire" he had no nationality and no loyalty to anyone.
In looking at the description of Pethor, Rashi says that "land of his people" actually refers to the original people of Balak and not Bil'am. He explains that Balak was originally a prince of Midian who had been asked to take over in Moav, because of the defeat of Moav by Sichon. In fact, Rashi says (from Medrash Tanchumah 4), that Bil'am had prophesied that Balak would become a king. Now that he was a king and needed help, he sent to his original land (where Bil'am was currently residing) for Bil'am to come to him. Since the Torah was referring to a location that was known to the people of that time, it did not need to be more precise. Note that "by the river of the land of his people" means that the river and the people were known and this was to say that it was this Pethor and not another of the same of similar name. For example, New York City is only city in the United States that does not have another city existing with the same name. That is why one needs to say "Odessa, Texas" or "Baltimore on the Chesapeake Bay". Here too, one has to say Pethor on the river of the land of his (Balak's) people. Since everyone knew where Balak came from, then the river was also known.