The tribes who initially settled in England, identified by Bede as the Jutes, Angles, and Saxons, weren't from what we would these days identify as Germany, but from Denmark and the low countries. In fact the most similar language to Old English is Frisian, a language spoken along the low countries nearing Denmark.

Now, the notion of countries as political and cultural entities was very different in the early middle ages, and they typically tended to lack a notion of 'nationalism' as they do today; cultural identity was more related to social structures such as family and tribe.

They certainly would not identify with the state of Germany or Denmark, but perhaps might with the language, for example, or similar cultural practice; the norse people identified themselves largely with their language, as Old Norse was relatively consistent in the Early Middle ages; from literary record it seems likely that an Icelander would have been able to understand a man from Gotland, etc.

There certainly was a certain attachment to the continental way of things; the House of Wessex, of which Alfred was a member, traced their lineage back to Woden, or their 'version' of Odin. The Sutton Hoo helmet is of a Swedish type, and Anglo-Saxon art is closely related to other 'Germanic' styles. In other words the English certainly did identify with German common culture, as the Germans and Scandinavians also did. They probably did not identify with tribal states.

The literary records show that the English were extremely conscious of the Germanic heritage; it was Alcuin, if memory serves, who chastised the English monastic tradition, asking @what does Ingeld have to do with Christ?". Ingeld, a character in Beowulf, is a 'German' cultural hero, in the same way that Siegfried/Sigurd is.

So, when Ottar/ Ohtere went to King Alfred's court, conversation of the type you are interested in would not have revolved around "Oh, my ancestors came from this patch of Denmark". Because cultural identity was not so related to where one came from, but who that person was; rather, an Anglo-Saxon thegn would have identified with Ottar's Germanic culture in general.

This is a rambling answer, so I'll do a quick recap:

Identity was culturally and lineage based

Nationhood was less geographically defined then it was today

More overlap would have occured in their cultural similarity (which only went so far; remember, the English converted quickly to Christianity and seemed to have forgotten their Germanic deities rather quickly)