I understand that the Byzantine Empire saw themselves as Romans, and that many Greeks living in the Aegean Sea considered themselves Romans as late as the 19th century. But when did the individual groups in the Western Roman Empire start seeing themselves (for example) as Franks and not Romans after the Western Roman Empire collapsed?
Aaah... my old nemesis question. I think I've been answering this question for close to six years now. If you dig around in my search history you'll find some of my other responses, but in the interest of pushing some new ideas of mine a little bit...
Something more interesting to think about, isn't "when did Romans stop seeing themselves as Romans?" Because the answer is "they never stopped" because the people currently living in the city of Rome still see themselves as Romans. Also, there is an entire region in Italy called "Romagna," so called because it was where the "Romans" (i.e. Byzantines) were based in the early medieval period (and different from Lazio), and if pressed for a question about local identity, they might say they're Romagnans. This is applicable because the southern aristocratic Gauls in Merovingian France were called "Romans" but it was seen as more of a geographic identifier than one with cultural affinity toward the empire of pax romana. Which means we have people here who SAY they're Romans, but may not mean what we think Romans should be (which is related to the earlier point about current residents of Rome being Romans).
The question you really want answered is "when did who we think of as Romans stop thinking of themselves as what we think of as Romans?"
Give that question a think for a bit.
This is quite difficult to answer on many levels, up to and including the fact that we can't always agree (and scholars certainly don't) on "what we mean when we say Roman," which is problematic both for identification within a specific period in the past, identification within a moving period in the past, and identification of a stop and start date within the past and the present.
tl;dr - So long as there are people in the city of Rome which still calls itself Rome, there'll still be people to see themselves as Romans.
Hey! While there's always room for more discussion, you might be interested in my previous answer on the transformation of identity late ancient and early medieval Gaul-Francia.