Edit: re-reading the title several hours later, I see that it looks like I could be asking “Of those three, which would they have been?” To clarify, I mean “Would they have been considered any of these (respective to where they particularly were)?”
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It doesn't come up often, but from time to time, my brother makes a comment about how we are part-Russian/Polish/Austrian on mom's side because that's where our great-grandparents came from, but it's never been clear to me if that's technically correct. I feel like I'm more Jewish on my mom's side than I am any of those three (i.e., it feels like we 'inherited' an ethnic religion, more than geographic heritage)—though it's possible that some more localized customs slipped away after a few generations in America.
I don't really have any personal details about their lives to share, so I can't get too specific an answer, but generally speaking: how would Jews living in the Pale of Settlement identify with their national borders, and did that change if/when they emigrated? How did non-Jews identify them?
By fellow Jews, very likely, though they'd have divided it up differently. While anecdote isn't a good enough source here on r/AskHistorians, my grandfather loves to tell of how when his mother first came to the Lower East Side in the 1920s and brought her kids to the park, the mothers there told her which benches were for the Galitzianers (Jews from the Galicia region of Austria-Hungary) and which were for the Hungarians and which were for the Poilishers...
I wrote a bit about inter-regional stereotypes among European Jews here- nationality would have only been a small part of it, as is clear when you see that the example I give of a vast divide is between two regions in Austria-Hungary! At a time of huge empires (when most Jews came to the US, the vast majority of Central/Eastern Europe was either ruled by Germany, Austria-Hungary or Russia), the specific kingdom was technically relevant in that it could have relevance on Jews' day to day lives and experiences (for example, Russian Jews were somewhat more likely to specifically have escaped antisemitism than Austro-Hungarian Jews, particularly in the 19th century) but more important internally was the specific region- or even the specific town! Landsmanshaften, or mutual aid societies to which belonged people who had emigrated from the same towns and villages, were key in helping new immigrants acculturate, providing cradle-to-grave support, sending money back home, and providing cultural and religious frameworks for immigrants.
In terms of non-Jewish Americans, the average non-Jewish American didn't necessarily care which particular country a Jewish immigrant came from and lumped them in as Jews. The US government, though, generally did. In general, pre-1920s nationality of immigrants only actually MATTERED if the immigrants were Asian (in that if they were Asian- and in the case of some particular laws, specifically Chinese or Japanese- they were at various points just not allowed in). Immigrants' countries of origin were noted but not necessarily of importance once they arrived. Starting in the 1920s, first with the Emergency Quota Act and then with the Johnson-Reed Act, in addition to solidifying restrictions/bans on Asian immigrants, quotas were put into place based on nationality, particularly based on the nationality of origin of current US residents. One of the main reasons for these laws, in addition to the restrictions on Asians, was to prevent Eastern and Southern Europeans from emigrating in the large numbers in which they'd been coming- including Jews, more than two million of whom had previously emigrated to the US from 1881-1924. After that Jewish immigration, among that of basically anyone who hadn't been born in the UK or Ireland (and to a lesser extent Germany), slowed to a trickle.
The question of where the immigrant's Judaism came into it is an interesting one, though! I just took a look at my great-grandmother's naturalization paperwork, for example- she emigrated in 1926 (an exception to the above trend!) and was naturalized in 1935, her country of origin was listed as Poland, and her race was listed as Hebrew! But if you want to blur the lines a bit more, another fascinating angle- my great-grandmother was able to emigrate in 1926 because her husband, before their marriage in Poland, had lived in the US for a decade and been naturalized himself. Therefore, when my great-grandmother came over with their two children, she was the only immigrant- her children were born US citizens. So it's absolutely fascinating to see, when you look at their ship manifest, that while my great-grandmother's race is listed as Hebrew, her children's races were listed as USA! Race designation in immigration in this period was pretty arbitrary, with race and ethnicity generally conflated and conflicting definitions of race to begin with, and it wasn't until the 1940s that "Hebrew" was taken off the list of races used for immigrants. (My assumption is that because my great-aunts were US citizens and not immigrants, they were not seen as needing to be classified, but honestly I'm not sure.)