A combination of several reasons.
It can admittedly be quite interesting for genealogical and family history reasons to figure out where your ancestors came from. However, claiming a set of national identities for oneself when that ancestry is several generations back, mixed in with other ancestry for several generations, AND you yourself is born and raised on another continent in another culture I find quite dubious. For example, in my family, there are kids that have one non-european parent of mixed “racial”(*) descent (let’s call it X). However, they both look like and are raised like other kids would be around here, plus they are served extra cultural tidbits from this parent. Awareness of where this parent comes from and getting to know his/her relatives is a good thing, but for the kids therefore claiming being X is quite dubious. As for myself, there is some mixing in from individuals from Germany and from Sweden several generations back, but for me claiming that I’m therefore somehow German or Swedish is plain ridiculous, both “blood-wise”(**) (for lack of a better word) and culturally. Genalogical interest is a different matter entirely.
Then you have in-groups vs. out-groups. These can be, but are not required to be, related to genetic similarities, family closeness or cultural similarities; anything that bonds people together goes. If you move far away to a new place, with very limited communication to your family and friends that you leave behind, you will in practice get a new in-group that you relate to and, perhaps, identify with (actually hopefully, as it would not be a good life with no in-group to identify with). Claiming to belong to the in-group that (some of) your great-grandparents – or even further away – is highly dubious, as belonging to a group goes both ways – the individual has to identify with the group, and the other members have to somehow identify with you.
And then you have the social and cultural stuff. If you’re born and raised on another continent, you are raised in and conditioned to the cultural and societal context of your local and larger community(***). Even if you are adopted from far away (call it X), you will normally be raised in and conditioned with the local culture and customs (let’s call it ξ). Thus, for all practical intents and purposes, you will be ξ, and X is something that will only matter later in life, if you for genealogical purposes want to know more about your biological family. Adopted kids (especially if they have significantly different genetic characteristics so that it is obvious that the kids are not biologically related to their adoptive parents) are admittedly special cases where the desire and drive for the pursuit of actual biological ancestry can be stronger than for non-adopted kids. But in the big picture, this is not really relevant here.
And lastly, there is the question of citizenship. To claim you’re being Scandinavian (Swedish/Danish/Norwegian), or Italian, or French, or whatever, implies actually having a citizenship (and can thus hold a passport) of said countries. If you say you are French, this implies that you are a citizen of the republic of France. If you don’t, then you are NOT French. However, this does not preclude you from living in France, have a French s.o., or having ancestors coming from France, or being fascinated with French culture and history. But without a citizenship, you are decidedly NOT French.
HOWEVER, doing as you do, @boomer47, specifying that you are of a particular descent and being aware of and interested in where your ancestors came from, is unproblematic. So if @mr.macabre had specified that there is some Scandinavian ancestry in his family, and stuck to that, it would be unproblematic. But he claims BEING Scandinavian (and I an highly dubious there, for all the above reasons) and even “viking” (which is just plain ridiculous).
Oh, and a later addition: There is also the purely semantic aspect here. Whenever I hear someone say that they “are X” or “come from X”, then I immediately associate this with either this person physically coming from place X (not his/her ancestors) or actually holding citizenship in country X.
This is by now getting too long, so I think I’ll stop there.
(*) Science agrees there is only one race of homo sapiens; individuals from all parts of the world can intermix and produce viable offspring. Thus, any genetic difference is due local variation mainly because of historical limits on mass communication, regional sexual preferences, and perhaps also xenophobia.
(**) The notion of being of descent Y “by blood” sounds strange to me. It’s not the blood that transmits the genes, but the gametes. Being X or Y “by blood” is a language construct that stems from olden days from before anyone had even thought about genetics and what actually happens with animal (h. sapiens included) reproduction, and from when blood was attributed more or less magical properties and from the observation that if you let too much blood out, life ceases to exist in the individual. The significance of blood is therefore a symbolical proxy of descent rather than actual.
(***) You can always argue that there are exceptions, like closed sects and closed communities where there is little or no cultural or genetical interchange with the larger society; Hasidic Jews are a good example.
But for the context of this post, they are irrelevant. ← Actually, strike that. On closer thought, I think Hasidic Jews are interesting examples of special cases that I think underline what I’m trying to convey. They are closed societies that do not interact easily with the larger community, and they are keeping language and cultural traditions and identities alive, so one can argue that they actually have a different identity than the rest of the surrounding society. Most other sects do not even come close to the Hasidic Jews in preserving the culture and language of their ancestors in another country, so THEY (the other sects) are more or less irrelevant.