What denominations do that? I'm familiar with Hebrew Roots, but doesn't seem to be what you're describing.
It wasn't until later that I learned that there is a second, much more popular movement under the name of Messianic Judaism which are people who are not ethnically or culturally Jewish who have determined that Christianity should return to its Jewish roots. These people have no historical connections to Judaism and usually grew up within a Christian cultural context. There is a lot of overlap with the "Hebrew Roots" movement that you mentioned, and in my opinion there isn't a real distinction between the two.
Myself I feel kind of biased but I view the first kind as more "legitimate" since Judaism, isn't merely a religion, it's a living, breathing culture and it is super weird for someone to just roll up and claim it without having any connection to anyone who was doing it before. It's like if I decided I was going to be Indian and started wearing stereotypical Indian traditional dress and eating only curry because I think that's what Indians eat, without having any actual Indians in my movement.
> It considers itself to be a form of Judaism but is generally considered to be a sect of Christianity,[2][3] including by all major groups within mainstream Judaism, since Jews consider belief in Jesus as the Messiah and divine in the form of God the Son (and the doctrine of the Trinity in general) to be among the most defining distinctions between Judaism and Christianity. It is also generally considered a Christian sect by scholars and other Christian groups.
It would be somewhat like saying you were a Messianic Christian because you believed that Mohammed was a later prophet. There is a word for that kind of religion, and it isn’t Christianity.
What it means to be a Jew is complicated. Jews form an ethnicity of interconnected people with a range of beliefs and practices (it is, definitionally, not whether one is religiously adherent to Judaism). To me, one could in principle be religiously Christian and also ethnically Jewish (that's an unusual view among Jews), but to do that requires having an actual connection to the Jewish ethnicity (e.g. if one was raised ethnically Jewish and maintains a Jewish identity). My impression is that "Messianic Jews" are religiously and ethnically Christian who are importing Jewish practices into their otherwise non-Jewish identity. If OP's wife was born Jewish or converted prior becoming a "Messianic Jew," I would stand corrected.
If I, a very white person, start singing songs from Back churches, that doesn't make me Black. I wouldn't face the real-world struggles against racism of Black people, for example, and I think that's a useful hint when thinking about who is and isn't a member of a minority group like Jews. Likewise, acting out Jewish practices doesn't necessarily make one a Jew, and as one example it doesn't subject one to the sorts of anti-Semitism faced by Jews. I'm not saying facing anti-Semitism a necessary or sufficient condition for being a Jew, but if not that, then there must be something else that connects one to the Jewish ethnicity --- the interconnected people who believe they are Jews --- other than just by saying so.