It is basically an adverb of an. Compare German eins "one", ein- "a, an*, quantifier einige "some, a couple, at least one", adverbial "Ich mache einiges". The negation of einige is not in use, doesn't exist, and would be ambiguous (nothing? many?).
Yet, not any means "nothing". Someone might figure the opposite of nothing were "everything", which is not necessarily the case, hence the confusion. Logically, the opposite is any, "there exists at least one" (this is very basic in logic). The negation of that is "not any" = "nothing".
I think that explains the semantic difference, but its hard to say when it appeared. I believe that would give a better answer than a strictly synchronic grammar.
Getting all the different variants together, "anyone can", "any one man can", "has anyone seen", ... would be a start. Comparing German to just these, we have "ein jederman", alternatively "jederman", "jeder" in the sense "everybody"; And "irgend-jemand", "irgendwer" in the sense "someone". For "je", which has a complicated history (confer Grimm), I find tempting to see it in the "y" in "anyone" (anyman??), but that's just an unsubstantiated hunch. Further we have "want any?", which would be "willst du welche", to which one can answer "ja, einige", though it would be rather "ja, ein paar" (a couple). Incidentally, "je" also translates to "ever" ("wenn ich je ..." - if I ever) in some context, and akin to "yet" eventually ("hast du je, jemals..." - have you ever, have you yet?) ...
The really funny one is the case where any is seemingly replacing a ~ an. Do you have any idea? At all? Germans would just say "hast du eine Idee?" Since we had this deranged thread today about the correct lambda expression for "Alice and Bob eat two cookies", that is necessarily ambiguous ... while I have no idea about that it appears intuitive to understand "have any idea" adverbail, not counting the objects, but the verb. This finds a parallel in "hast du mal einen Stift". "mal" is also hard to explain, even for native speakers, e.g. in du kannst schon mal auspacken. Since "schon" translates most often to "yet" ("hast du schon" - do you have yet?), and "schon mal" is a colocation, it may be reasonable to compare "mal" to "je", "jemals", "yet", "ever", circling back to the above argument.
There's a dense cluster of multiple parallel developments. More examples would be required, preferably archaic ones. Preferably about "any" itself.
Finally, "einige" is not to be confused with "einig" (united, in agreement) or even "geeinigt", "vereint" (united), though "uni" might be interesting to compare. Since the -ig appeared hard for me to explain for a second, I came to remember "einje", which is common in northern dialects, at least Berlin and surroundings. That would be much closer to "any", phonetically. That might be just a common development. Or it might imply that "einige" was a backformation from "einje", which would be a composition of "ein" and "je". Just which "je", of which there were many, I don't know. We also say "einwas" (an what; sounds like an ones without nasalisation), so make of that what you will.