This is a great question with a complicated answer.

On the one hand, the Anglo-Saxons were technically at least as much Romano-British as they were Germanic. Isotope studies of migration age skeletons' teeth, and a recent survey of English DNA, suggest that the actual agress from Germany into Britain was small-scale, and that the immigrants blended into the larger Romano-British population. Culturally, 'Anglo-Saxon' archaeological remains show a deliberate blending British and Germanic practices that does not suggest the cultural dominance of German immigrants - it shows, instead, a people creatively reimagining itself by drawing on diverse sets of cultural symbols and resources. So rather than thinking of Anglo-Saxon as kingdoms of Germanic immigrants, it's more accurate to imagine a bunch of people from all over the Roman world (and its fringes), but mostly from different parts of Britain, rebuilding a new society within the ruins left by the collapse of Roman economic systems and industrial infrastructure.

In ninth century politics, however, stories of Germanic origins had become very important. Various leaders played up or invented stories of their descent from Angle or Saxon immigrants, and histories (like Bede's) and poems (like Beowulf) were being written which emphasized the Germanic origin story and de-emphasized the role of non-immigrant ancestors in Britain's secular and church history. So while it's not really true that the Anglo-Saxons were actually a bunch of German immigrants, that is the story many Anglo-Saxons were telling about themselves.

But that was not in itself enough to guarantee that the Anglo-Saxons would consider themselves to be related to the northmen. While it must have been obvious that Norse and Old English were similar languages, the idea that shared languages implies shared kinship and culture is a product of 19th century anthropology and scientific racism. The ninth century Anglo-Saxons imagined themselves fitting into a set of social and kinship relations that weren't based on language, genetics, or immigration; while descent from Saxon ancestors was an important part of many English kings' political identities, it did not decine their contemporary community. Instead, this identity was based upon the Anglo-Saxons' relationship to Christianity and the Roman church. Historians like Bede argued that the tribe the Anglo-Saxons were most closely related to was Israel, because of their providential conversion to Christianity; their Saxon ancestry was simply a waypoint on their journey toward their true identity as God's chosen Christian people. Saxon origins weren't a fact of their blood that continued to shape their future (the idea that Germanic ancestry governs a people's destiny belongs to the 20th century, not Anglo-Saxon England); it was just a part of their past.

So on the one hand, the Anglo-Saxons weren't exactly descended from German immigrants (at least not in a simple or significant way); but they did imagine themselves as coming out of Saxony; but that imagined past was not, in itself, enough to make them feel kinship with the pagan pirates and slavers who started attacking their churches from the sea. Some Anglo-Saxons may have remembered how (according to Bede) their pagan ancestors had been sent by God as a judgement on Britain's sin, and perhaps these new pagan enemies were also a judgment from God. But that didn't make them kin - far from it!