No, it's not a plural marking. Both words are adverbs and they both mean the same thing (except "anyways" is only used in some dialects). English doesn't have plural adverbs. The "s", in essence, is meaningless.
Specifically, the word is used in some American English dialects. One site says:
The Merriam-Webster Dictionary of English Usage cites the Dictionary of American Regional English to note that anyways is apparently strongest in the South and South Midland (both U.S.) dialect regions.
However, it is much more widespread than those two regions, since it is used in my dialect, which is North Midland.
Etymologically, according to the OED, the word was originally written "eanies weis", with both "any" and "way" being written in the genitive singular case (this was in early Middle English). A little later (still in Middle English), only "way" was written with the genitive singular ending, with "any" just being written normal.
You can see this same -s ending on forward(s) and backward(s). Although it doesn't look it (because the spelling and sometimes pronunciation has changed), it's also the basis for the ending of words like amid(st), and also words like once and then(ce) (see also here).
(You don't need to know the etymology to use the word or any of the other words I mentioned, so don't let this section confuse you.)