Is there a common origin? No. Is there some theory to explain this? I propose one: common need.
In Is “Huh?” a Universal Word? Conversational Infrastructure and the Convergent Evolution of Linguistic Items, Dingemanse et. al. have found that in 10 languages, (and less carefully studied, 30 languages)
Huh? is universal, and that it is a word. ...the similarities in form and function of this interjection across languages are much greater than expected by chance. ... it is a lexical, conventionalised form that has to be learnt, unlike grunts or emotional cries. We discuss possible reasons for the cross-linguistic similarity and propose an account in terms of convergent evolution. Huh? is a universal word ...because it is shaped by selective pressures in an interactional environment that all languages share: that of other-initiated repair.
I would propose a similar mechanism for "so-so": although it is not the same word in every language, there is a strong need in language for a short, meaningful expression that connotes the ambiguity of so so.
I would add that in French, the expression is comme ci comme ça, Spanish asi asi (or mas o menos), in Finnish niin ja näin, etc.
Edited to add: why would the word/phrase be echoic? So-so is a short phrase that truly admits to the bad existing with the good in approximately equal measure. It's a hard thing to talk about ("Well, he's in the ICU in a coma on a respirator, but the neurologists tell us there's still a fair amount of brain activity, and no evidence of increasing intracranial pressure, so it looks like he might come out of this, but we don't know how intact he'll be."). How's your marriage? So-so. How are your job prospects doing? So-so. Sure, there are minor events that could be applied to. But the principle is the same. If someone wants to be honest but doesn't want to elaborate, so-so indicated the ambiguity: some good, some bad. Like this, like that (French, Italian and Spanish).
This is how I understand it.