Does the search engine for the site find variants of the search words like plurals and possessives?

No it does not.

The search engine finds exact matches only.  It does not do what some people call "fuzzy searching".

So if you WANT to find word variants, type the variants in the search box.  Here is an example:

Instead of just typing mother, type

mother mothers mother's

You might want to even expand that by adding adjectival forms of the word as with

mother mothers mother's maternal