This doesn't seem to be much to do with communism. You can operate not for profits or charities in capitalist systems (in fact some of its strongest proponents would much prefer people give money directly to charities to help people, rather than pay taxes to a government that might help people).
I don't see what's communist about it. It's just that the developer has arrived at "charity" via the route of communism. The biggest societal difference is that chess.com allows people to play chess for a living, whereas a free platform does not.
That's why I noted "Thibault Communism" to narrow it to his specific thinking on this, which IMO is quite lovely and unique.
TD is the one who who articulated it this way, and there are pretty strong logical and operative links between his ideas and actions.
There are lots of differences between chess.com and Lichess, societal and otherwise.
Anyway, I'm not claiming that Lichess is superior in every way, just that it's great and unique. There are lots of successful companies/startups/sites operating on the chess.com mindset. Lichess is quite unique.
I don't see what's communist about it. It's just that the developer has arrived at "charity" via the route of communism. The biggest societal difference is that chess.com allows people to play chess for a living, whereas a free platform does not.