Sure, it's possible, but whichever app owns the database ultimately controls the data, the schema, etc. The file system provides a neutral database that all apps can cooperate within.

I guess what really matter is ultimate ownership of the data, if it's a sqlite-like db or a bunch of markdown files in my machine I can work with them, but if it's on a cloud (someone else's computer), then I'm doomed.

Why does any app have to "own" the database? I don't see this as being a restriction any more than it is with Markdown. Arguably even less so with a database since you have access to transactions.

If you made a new app what is "the" database you would write to? There isn't an existing standard for this aside from the file system.

That's a fair point in the sense that you're always guaranteed to have some file system already :)