As N. Post says, the use of "some" rather than "somewhat" in this question was probably influenced by the use of a poll format with other options such as "not too much," "not at all," and "a great deal" which don't have separate adverbial forms.
J O S H, kayleeFrye and Rathony mention the use of "some" as a general substitute for "somewhat."
However, I think for some people, there is something about the verb "care about" specifically that makes "care some about" sound more natural than it otherwise would. I grew up in California; when I read the title of your sentence I couldn't figure out what the problem was supposed to be. After reading the answers, I agree that "somewhat" sounds better in this context, but the version with "some" doesn't sound ungrammatical to me.
On the other hand, I think "I guess I like him some" (the Collins example that J O S H mentioned) sounds more noticeably colloquial to me.
Another thing that seems related is the use in negative sentences of "any" without any following word, in place of "at all." To me, this seems kind of possible, but marginal; I feel like I might try to use "any" in such a sentence, but when I look at it or hear it I feel uneasy.
You can find many examples of "don't care any about" online using Google search, but the sequence "care any about" doesn't show up in a search of the Google Ngram Viewer. This indicates that it occurred in less than 40 books in Google's corpus of scanned books published earlier than 2000. This seems to indicate that this use of "any" is confined to colloquial registers of speech, and is not accepted by copyeditors and other gatekeepers of "accepted" grammar.
There are some examples of "care any about" from books published after 2000 in Google Books. However, most of them seem to be used in dialogue or deliberately colloquial passages.