From my poli-sci background "Ethnicity" is an academic mainly anthropological refinement of the more generic term "folk".
As empires expanded many "folk" became Roman, or British, or Nordic nationality etc.... but were quite different from the founding empire.
While no one has made a hard and fast definition of "folk" anthropologists are continually defining and re-defining "ethnicity" using on-the-ground long-term research. Ethnicity is one of those terms like "race" "gender" and "class" that will not remain static but will continually change over time.
Ethnicity began from the roots of the same words as "pagan" and "heathen" and was originally used to refer to a race-based minority within a dominant class. However it has developed and continues to be re-defined to include elements of culture, language, religion etc....
This is the best online definition I could find:
In everyday language, the word ethnicity still has a ring of "minority issues" and "race relations", but in social anthropology, it refers to aspects of relationships between groups which consider themselves, and are regarded by others, as being culturally distinctive. Although it is true that "the discourse concerning ethnicity tends to concern itself with subnational units, or minorities of some kind or another" (Chapman et al., 1989: 17), majorities and dominant peoples are no less "ethnic" than minorities.
From the same passage:
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Through its dependence on long-term fieldwork, anthropology has the advantage of generating first-hand knowledge of social life at the level of everyday interaction. To a great extent, this is the locus where ethnicity is created and re-created. Ethnicity emerges and is made relevant through ongoing social situations and encounters, and through people's ways of coping with the demands and challenges of life. From its vantage-point right at the centre of local life, social anthropology is in a unique position to investigate these processes. Anthropological approaches also enable us to explore the ways in which ethnic relations are being defined and perceived by people; how they talk and think about their own group as well as other groups, and how particular world-views are being maintained or contested. The significance of ethnic membership to people can best be investigated through that detailed on-the-ground research which is the hallmark of anthropology. Finally, social anthropology, being a comparative discipline, studies both differences and similarities between ethnic phenomena. It thereby provides a nuanced and complex vision of ethnicity in the contemporary world.
The original question was:
Now, I am a follower of Asatru or Heathenism an old Germanic Northern European religion. However, it wasn't just a religion it was also a culture. The Vikings came out of this religion or cultural group of people and it colonized much of Europe during the Viking Age. However, I was wondering under this definition if the Viking people now re-created as Asatru or Heathenism would be called an Ethnic group with race as Caucasian or as a cultural group?
I'm not sure if you are asking:
a) Are the Astru a historical ethnic goup?
To which I believe the answer is yes.
b) Are the Vikings a historical ethnic group that would be a post-Astru ethnic group?
To which I believe the answer is yes.
c) Are contemporary people who self-identify as Astru and see a lineage to Viking history an ethnic group or a cultural group?
I think I would answer that: Currently they are a cultural group with the potential to become an ethnic classification.
I see cultural as something that has the potential to transcend history, we can identify Roman and Egyptian "culture" that was the "culture" a thousand and two thousand years ago without transcribing that culture onto today's Romans and Egyptians which now have a quite different and distinct culture to the historical culture.
Cultural markers can be studied and made tangiable through a careful study of anthropological history.
However ethnicity seems more contemporary and is not only how one self-identifies but also how one is outwardly identifiable.
In that way an ancient culture has the potential to become an ethnicity if enough people choose to revive that culture in contemporary times. Which seems to be what the Celtic/Gaulish/Irish are attempting to do.