One of the most difficult things that students of prehistoric and protohistoric cultures in the western portions of North America have to confront is the relationship of political entities to linguistic or other classifications. In most of California, and all of the NW coast south of the Tlingit and Haida areas, there was no political organization above household. Rich men had more influence. There may have been a shaman or doctor that had influence over a larger population but it was not formal. In addition, there may also have been religious leaders, often called dance leaders, that had influence that extended beyond the household. But the concept of tribe itself has virtually no utility.
Early anthropologists recognized that tribes were a construct invented by anthropologists to compare and contrast cultural traits (see P. E. Goddard Life and Culture of the Hupa 1903), but investigations at the scale of extended family just were not feasible. They therefore focused on the larger ethnolinguistic unit - using groups of people with homogeneous or mostly homogenous languages. This was in part because of the linguistic training of early anthropologists, but mostly because it was a seemingly reasonable way to divide populations up.
The problem is perhaps most clearly exemplified by the far Northwestern California and far Southwestern Oregon groups. This is an area where virtually each river drainage was occupied by a different language group from very different language families yet the cultures were virtually identical. Compare, for example NW California where the Tolowa (Athabascan) are bounded to the south by the Yurok (Algonkian Algic), to the east by Karuk (Hokan), to the Southeast by other Athabaskans (Hupa, and its subtribes or tribelets, the Chilula, Whilkut, and Tsnungwe), but these Athabaskans were unintelligible to the Tolowa. Further south are other Athabaskan groups that were not intelligible to any of the others and a variety of other ethnolinguistic groups. All these groups shared many dances, regalia, mythologies and other traits.
This area has been the site of extensive ethnological research beginning in 1871 and marked by fairly monumental research projects by Kroeber (see his work on the Yurok) and his students. While these "tribes" held many unique traits, having to do with aspects of religion, and customs, they were, for all intents and purposes, identical cultures. Complicating matters is the fact that Northern California natives were notorious polyglots. Powers once noted an elderly informant that "had one eye and six languages in his head".
So how relevant were the distinctions to the native groups themselves? In the area I study, I don't think it was very relevant. As an example, almost all natives that lived near boundaries of linguistic groups were bilingual so intercourse of both an economic and social nature was unimpeded between groups. People routinely married outside their linguistic groups. In fact, certain villages held alliances with other villages in different language areas. Alternatively, villages within the same language group were traditional enemies.
Many groups that have been identified as subtribes or tribelets of the Hupa, because they spoke dialects of the Hupa language felt no affiliation with the Hupa proper, in fact they felt the opposite. So my conclusion is that it probably made a little difference in that the natives recognized that there were other people that speak our language, but ethnologically and to the natives it was largely irrelevant.
See:
Handbook of California Indians, A. Kroeber 1970 (originally published 1926)
California Indian Languages, Victor Golla, 2011
Life and Culture of the Hupa, P.E. Goddard, 1903
California Athabaskan Groups, Martin Baumhoff 1958.
Edit: economic for economical Edit II: Algic for Algonkian
I'm late to this party, but I wanted to chime in with a few points. Namely that the two examples you give, the Cherokee and the Choctaw, have some significant sociopolitical structures backing up those labels. They're not just names given to them by the colonists. In both cases, there were national councils developed to unify the Tsalagi and Chahta peoples, respectively. Of course, in both cases, there were regional distinctions, such as localized dialects and slight cultural differences, but that's true of all sufficiently large nations. However, the differences between these internal groups were far less than, say, the differences between the Five Nations of the Haudenosaunee.
Linguistically, the Choctaw recognized that they were closely tied to the Chickasaw, long before any Euroamerican linguistic lumped them both into the Muskogean languages. Their ties to the Chickasaw are commemorated in a shared origins narrative, explaining how the two nations split over a disagreement concerning where to settle once they crossed the Mississippi from the west. For the Cherokee in the colonial period, linguistic connections with their neighbors were less important, mainly because there were far fewer neighbors that spoke a similar language. The Haudenosaunee, however, eventually came to see the Cherokee as distant cousins due to linguistic and cultural similarities, and - once they stopped fighting over Ohio - they engaged in some cultural exchange. We have, for example, the account of a Seneca traveler among the Cherokee, who went south to see what the Cherokee where up to.