I think you're confusing events here. The event in 1066 that is (sometimes, not always) considered a bookend of the Viking Age is not the Norman invasion and battle of Hastings, but the event a few weeks earlier; the defeat of Harald Hardrada and the Norwegians at the Battle of Stamford Bridge, which by-and-large would mark the end of Viking rule and invasions in England, a series of events that started at Lindesfarne. The Norse origins of the Norman ruling class doesn't play into it.

These are not hard dates; Scandinavians went abroad on similar Viking journeys well prior to the Viking Age in the Baltic region; as seen with the Salme ship finds in Estonia for instance. However above all you have the domestic cultural shifts at the beginning and end of the Viking Age, the end of course being primarily marked by the establishment of Christianity. That is not something that proceeded equally fast everywhere in Scandinavia; by 1066 Norway, Denmark and the Geats were largely Christian but there was substantial resistance in Sweden proper, in particular the Mälar valley region until the early 12th century. The Viking Age therefore does not have fixed dates; it's often rounded off to 800-1050. Sometimes it's extended to 1100 if christening of Svealand is to be included.

As for the Normans they did not consider themselves French in the present-day sense and the Scandinavians did not consider themselves to be vikings. Vikings were vikings. Scandinavians were Danes, Norwegians, Geats and Swedes and subdivisions thereof. A viking was someone who went off on journeys to trade, raid, engage in piracy or other things, and not their ethnic identity. Non-Scandinavian pirates and such could also be referred to as vikings. Being Christian is not necessarily at odds with being a viking either; many vikings were Christian; for instance most of the almost 30 runestone memorializing those dead on the ill-fated journey of Ingvar the Far-Travelled are explicitly Christian. If anything, vikings were more likely to be Christians as it was this traveled class which'd come in contact with the and decided to build private chapels and such, after returning home with wealth and new ideas. One such example is Kata Gård in Västergötland, Sweden, where a chapel was built by a powerful family a century before most of the country had become Christian, but people there continued to go on viking journeys; nearby runestones (and Anglo-Saxon coins) bear witness of locals who followed Cnut the Great to England. (1015)

So while the Viking Age is a concept in Scandinavian historigraphy, the term vikings as a catch-all demonym for Scandiavians in Scandinavia during the Viking Age is not used much in serious contexts (museum marketing to tourists is another matter; as is describing them abroad where they were often a mixture of nationalities and since they'd traveled they were by definition vikings. Norse is a better term but even then should be recognized that the term is a grouping of multiple nationalities)

But the Normans are generally not considered Norse, nor are the Rus' or Goths or other groups which may or may not have originated in Iron Age Scandinavia. In the former two cases the Norse component was only ever a small ruling elite which quickly assimilated, in the latter case they may not have even originated in Scandinavia at all.