Noted Communist Thomas Jefferson proposed it rather before the Communist Manifesto was written (not that he came up with the idea, of course), and some US states and various other places started introducing it from the early 19th century on.

Just because Marx was in favour of something doesn’t necessarily make it inherently socialist.

But something can be socialist even if it came about before the idea of socialism was put forth.

Remote villages have lived in communistic systems but have no idea about the concept of communism.

Are you using ‘socialism’ to mean ‘doing anything for the common good’? I mean, free education seems completely orthogonal to socialism as normally defined.

How so? Please give me your definition of socialism and reasoning for calling it "orthogonal" to free education.

Not the OP, but Socialism is generally concerned with democratic ownership of the means of production.

Like stock options for employees?

Only if the company doesn’t have any other shareholders. I think John Lewis is run like that? Devil is in the details, and I’ve never had reason to look closely in their case: https://en.m.wikipedia.org/wiki/John_Lewis_Partnership

I would argue that “things in the Communist Manifesto” defines communism, in the same way that “things in the Bible” defines Christianity even though some (lots?) of the Bible originated in Judaism.

That said, I was asking for an example of the concept of free schooling from before socialism, and I got it :)