Nope, Hungarians definitely do not self-identify with Slavs. No way, no how. But they are ethnically Slavs. It's like the English claiming to be Germanic and not the same people as Irish, although the Magyar are far more Slavic than the English are Germanic.
My mother is 100% ethnically Hungarian and I've tested her and one of her brothers at all the companies and Gedmatch. They are overwhelmingly Slavic. They have a tiny percentage of Turkish and Mongolian but so do many other East Europeans.
They do have a small percentage of Finnish and Estonian and many, many 8 to 12 cM matches in Finnland. My mtDNA has 0 difference with several people of both East European and Finnish descent. However, there are no Finns in the family tree. I can trace their family back 400-500 years in the same little cluster of villages in the northeast-Hungary: Bezded, Zahony, etc. This may be crazy, but I chalk that up to the Magyar, who originally came from the forests north of the Ural mountains and are of the same language group as the Estonians and Finns.
However, except linguistically, the Magyar was absorbed by the Slavic population. My Hungarian ancestors are very, very similar culturally to Ukrainians. But identity-wise, there is no way they would call themselves Slavic.