Arguably, since “anything” is only used in negative sentences, it is kind of part of the negative. You wouldn’t say “I have anything”, only “I don’t have anything”. Then again, it can be used in positive sentences like “I would do anything” with a slightly different meaning.
But let’s take “anymore”. “I can’t stand it anymore” is a common phrase but “I can stand it anymore” not so much. “anymore” is only used in negative sentences so the “not” is arguably redundant and therefore “not … anymore” is kind of a double negative.
And even that’s not true for all speakers of English. There are native speakers who would say “I go to town anymore” unironically.
Edit: I reread the comment you reacted to and to say it’s “to emphasize” is wrong. That argument would work better for “at all” for example. My point is that “double negative” isn’t as clear cut as it might seem to be at first glance. “I don’t know nothing” is a double negative for sure, “I don’t know anything at all” kind of in a way, “I don’t know anything” maybe a little bit.