Does this word sound wrong, uneducated, perhaps some recently introduced error? In fact, it’s older than its pair anyway, but only slightly. The s is not a plural; like many s’s that show up in odd places, and notably at the ends of family names (Banks, Woods, etc.), it’s a genitive, which is to say a possessive. In this case it’s an adverbial genitive, so meaning “of or by any way.” Compare always, besides, etc. There is no apostrophe because apostrophes on genitives are a newer invention. And the active uses of genitives have reduced somewhat in modern English.
But anyways! None of the preceding changes the fact that for many hearers anyways is slangier, more casual, and so on, and we have to proceed accordingly. Which way to proceed? The multiple ways available can be seen in the y’s, forking like Frost’s roads (though some might see them as the drain they insist the language is going down). In between they are echoed, truncated, by the w. A separate echo effect comes from the two a’s. Ironically for all that, the sounds aren’t repeated: the first a is a lower and laxer vowel than the second, and not a diphthong as the second is either. And the second y is part of that diphthong, whereas the first is [i] (or, for some speakers, more of a lax central vowel). Cap it all off with an alveolar buzz, and you will notice that the whole word is voiced. Not only that, it all happens towards the front of the mouth – except for the raised back of the tongue in the w. So there aren’t so many ways in anyways that you make sounds.
And whence come this compound’s parts? Germanic: the any comes from the same source as one (which one? any one), and way, referring first to a road or path, comes ultimately from an Indo-European root that has evolved to words in a variety of languages from Sanskrit to Icelandic for routes, travel, transporting, and vehicles.