And yes, you probably should, if only to pre-register your ownership thereof if google ever decides to nuke you from orbit

Google's way of tying real identifies of people to domains, without making it explicit.

Basically, your domain will be weirdly treated by a bunch of entities, none the less Google themselves, if you don't add your domain there (or some other Google property).

Especially with less common TLDs, like .online, they really want to be able to tie it to some identity, so unless you add it there, eventually your domain ends up on some sort of blacklist, in the case of the author it seems they used the "Google Safe Browsing" blacklist to get the author to involve Google somehow.

But if you do - you would get some notifications from Google about that website/domain.

I've only ever seen emails of the "There's an increase in 4xx/5xx errors on site/page(s)"

Was called webmastertools before.

You can also request Google to index your site on GSC as well.

You should probably add your websites to GSC.