The way in which cultures and groups deal with situating themselves in the world is remarkably varied depending on the culture, and this is just as true for North America as it is for any other location. The short answer is that yes, Native Americans, First Nations, Inuit, all had names for places, but not like we have "Germany, America, China" and so on.

It should be somewhat obvious that anyone inhabiting an area will have ways of situating him/herself on it through language. For example, in the north, each language group had well-known names for thousands of lakes, rivers, streams, portages, villages, bays, as well as landmarks such as slides, and the locations of various historical events (going back several thousand years). In addition to this, areas could be referred to by the name of the family/group/band working there, as in "so-n-so's area", and these are strategies that were likely used by almost all First Nations / Native Americans.

In much of the Pacific North West, including British Columbia, territory was owned by communities, by individuals, and often attached to names and stories, with "places" or areas potentially changing ownership through the transfer of names, or through marriages. In Bella Coola where I live now, each mountain has a name, stemming from a story, and that story is linked to the origin story of the nation, often of specific families, with specific individuals sometimes bearing the names of landmarks and characters associated with them. The language likewise has names for other nations extended for thousands of km in all directions, names for their villages, languages and territories.

Now to the next part. You present your question as an either or, but in reality it's not that simple. Ideas such as those of the modern geographically defined nation state and the concept of citizenship are products of a political world-view that defines individuals in terms of their relationship to government structure - your status, your rights as an individual are legally determined between you and the state. While this idea of primary identity being determined through a legal relationship between a government and a subject might by partially correct for the central and south american empires such as those of the Aztecs, Ina, or the Mayans, this way of viewing identity is not a useful approach for most other groups in NA historically. Rather than seeing themselves as a "German, with allegiance to the German Government", a Nuxalk or Gitxsan person would see themselves as a member of their family/crest (respectively), with obligations to communities and individuals rather than to a nation. And of course identity based on these connections always held precedent over geography or place of birth - no one was considered to be Nuxalk just for being born in Nuxalk, identity came alternatively through something like family/crest/house/community connections and adoption (most of the pacific northwest), or through language (one of the most common Cree sayings translates roughly "of course I'm a Cree, as I speak Cree"), or through lifestyle (I've heard a few Dene Sulhine say that to live as a Dene is to be a Dene), or through a combination of the above. Location was important as stories and culture connected a culture to land and gave it meaning, but the very Platonistic concepts of the nation-state and the western views on separation of land from story, and the human from culture had little bearing on the various Native American nations.

I could go on and on, about how languages divided up land in unique and fascinating ways, but I think I'll stop there unless you have further questions.

Coast Salish communities in the Pacific Northwest had (and still have) both specific names to refer to themselves and outsiders, as well as place names that reveal important things about how their histories connect to land.

For instance, a person belonging to one of these communities in what is now the Fraser Valley area of BC might have identified as Xwélmexw (human beings/People of Life), an identity related to their kinship network and language. You might also identify yourself with the name of your village, family, and certain aspects of your spiritual identity. Salish people call(ed) other Indigenous people who did not share their cultural background lats'umexw (different). When Salish peoples met explorers and settlers, they called them Xwélitem (the hungry ones - a reference to their lack of food, and the fact that Salish peoples often fed them, but contemporarily this is also used more figuratively, to refer to the way settlers have appropriated Salish lands). For more on Coast Salish identities, see the work of Wayne Suttles (Coast Salish Essays), Crisca Bierwert (Brushed by Cedar, Living by the River), and Keith Thor Carlson (The Power of Place, the Problem of Time).

Similarly, place names dot the Salish landscape. These do not describe physical territories as much, though the general term among Stó:lō would be temexw, simply meaning land, earth, or territory. Rather, they explain how particular parts of the Salish world are deeply important to the history of the people who live there. There are at least two books on this, the excellent Coast-Salish Historical Atlas (Keith Thor Carlson, ed.), and Nooksack Place Names: Geography, Culture, Language (Allan Richardson and Brent Galloway).

Some excellent reading on this, if you're interested in the significance that place names have for contemporary Indigenous peoples, is the late (great) Keith Basso's book Wisdom Sits in Places: Landscape and Language Among the Western Apache.

Edit: sources added. Edit: formatting.