Origin of plurals and possessives

Solution 1:

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'.