So this question gets at two aspects of Anglo-Saxon identity, did medieval people attribute ethnic ancestry based on kingdom, and how reflective of the actual genetic make up of the population were these attributions?

The first is relatively straight forward to answer. Bede, our one "reliable" literary source for the earliest middle ages in England, more or less lays out for his readers where each of the tribes that came to England settled. The Jutes settled in Kent, the Saxons in the rest of the south, think Wessex, Sussex, Essex, and the Anglians settled in Mercia, East Anglia, and so on. I forget who got Northumbria truth be told. So that's relatively easy. More or less, certain kingdoms were believed to have been the product of certain tribes on the continent.

Was this reflected in day to day interactions between these various kingdoms?

Harder to say. We have very few written sources from this time frame and those that do survive do no capture the voice of every day lived experience, rather grander narratives of Christianizing, Wars between Bretwaldas and petty kings, and invasions. Archaeology cannot really inform us heavily on thought patterns or cultural stereotypes that may have predominated among the various kingdoms. However it is reasonably well established that these various kingdoms had at least an imagined ethnic coherency to them that resulted from their settlement/conquest by a continental analogue.

Furthermore, even the supposed ethnic antecedents of these kingdoms were significantly muddied by a few centuries of conquests. The Isle of Wight was settled by Jutes supposedly, but then conquered (and according to Bede ethnically cleansed) by Wessex. Wessex also grew to encompass Sussex and Essex, Kent, and Mercia, and eventually all of England, but had at various points been subjugated by the supposed Anglian kingdoms of East Anglia and Mercia, as well as the Jutish kingdom of Kent.

So how does this supposed match up of kingdoms and ethnicity match up to the actual events going on in England? Not very well of course not, if my use of the word imagined above didn't tip you off. Migration is generally not a simple or straightforward issue and it was no different in England. Archaeology has shown us that continental analogues for English settlements exist not only in Northern Germany and Jutland (modern Denmark) but also Norway and Sweden. Indeed the burials at Sutton Hoo in East Anglia have been shown to be remarkably similar to material culture in use in Southern Sweden. But the southern Swedes, or Geats, go unmentioned as sources of migrators by Bede.

More recent efforts at DNA mapping have shown, if not outright and definitively, that continental DNA is more prevalent on England's eastern coast than elsewhere in the country. While these results are not conclusive, and critics have pointed to later migration movements as having muddied the results, they do seem to point towards relatively heavy levels of migration from the continent to England. However, this still leaves 70%+ as natives who assimilated into these new identities. Even in these kingdoms dominated by supposed Anglians, Jutes, and Saxons though the new comers were not the majority.

The later Danelaw confuses these issues as well. It gets even more confusing after the Danelaw was conquered by the English, but England was then in turn conquered by the Danes outright. While there is evidence of a separate Danish identity clinging on in the Danelaw and after Svein and Knut's conquests, following the Norman conquest it was rapidly extinguished, probably not least because Northumbria was hard hit by the Harrying of the North.