What I'm referring to is the era when the citizens of these new Kingdoms stop differentiating themselves ethnically along the lines of either "Roman" or [Insert name of Germanic tribe].
We know that when the "Barbarian" invaders settled on what were previously Roman provinces, they didn't immediately intermingle with the local Romans. If I recall correctly, the Visigothic and the Frankish Kingdoms had different laws and punishments for people depending on their ethnicity based on either Roman Law or Germanic Customs.
So when did the citizens of these Kingdoms start to integrate into a society where the line between "Barbarian" and "Roman" blur, and eventually cease to exist altogether?
You might be interested by this old post, which, though it is centered on Aquitaine, tries to address the same question for Gaul. I can (try to) answer follow-up questions if you have any.
It really depends on "who" you're asking. During the First Crusade Anna Comnena -- daughter of Alexius I of Byzantium -- frequently referred to the Frankish peoples passing through the Eastern Roman Empire en route to Jerusalem as "barbarians" in the Alexiad (an account of her father's reign).
We can also see even more recent accounts of people describing Western Europeans in "savage" terms, albeit Muslim ones. Baha al-Din not only referred to them as barabarians, but also polytheists (the Trinity was not something that Muslims took kindly to) in The Rare and Wonderful Life of Saladin.
Beyond that, you have more local concepts of "barbarians". The Scots were frequently depicted as barbarians throughout the High Middle Ages. The famous line, "Who would deny that the Scots are barbarians?" is from an eyewitness account of the Siege of Lisbon in the late 12th century. I have a feeling that this depiction of the Scottish people didn't stop then, but that's where my knowledge ends.
So, as I said, it really depends on the region you're looking at for when the concept of Western Europeans stopped being considered "barbarians". It's impossible to really know, though, when the sentiment finally became extinct completely with regards to European peoples.