Historical linguistics emerged only around the 17th century, so no; it unlikely they discussed their common linguistic origins.

However, most if not all historical linguists believe that the people now known as Anglo-Saxons and Norse (or Danes as they were known contemporarily) could still communicate with each other to an extensive degree during much of the Viking Age.

The Anglo-Saxons themselves were also very much aware that, while they considered the Danes to be invaders, they themselves had also come over the sea and had driven back the indigenous Celts only some centuries earlier. In this regard, they would have almost certainly known that they were related to the Danes.

It is questionable however, to which extent they truly considered themselves to be related in a sociological sense, as whether someone was pagan or christian was of a much greater importance to the Anglo-Saxons than speaking a related language.