What is the origin of English plurals and possessives? English plurals look more French plurals, but I am not sure that is where they come from. As for possessives, I don't know where they come from.
The short answer is that they fell together when most of the other inflections were lost.
For the plural, many, many Germanic and Indo-European inflections used -s; it was one category of plural noun marker in just about every IE language. It was the only marker left when the others were elided.
Possessive '-s', on the other hand, has a very checkered history, which I won't touch.
As to the superficially related third person singular verb inflection, the '-s' is a Scandinavian loan, replacing original OE '-th', which came via Grimm's Law from PIE third person singular '-t'.