Had a thought, but some quick searching didn’t really give me much.
Is there such a thing as a browser that respects privacy, and can be synched through self hosted means?
I’d be looking for tab sync, bookmark, history, etc.
I use the floccus extension with Nextcloud as a backend for bookmarks/tabs and wallabag for read-it-later
Does it sync browsing history?
It can track a bookmark last visited time and count a bookmark number times xlicjed. It is a bookmark sync centric service
Brave sync server is open source and self host able.
Everything a browser syncs is syncable passwords, history, bookmarks, cards etc
The “issue” is there is not a user interface element to easily add the self hosted instance url
There are workarounds though
You can read my quick how to here
The Arch Wiki used to have some elaborate tricks on Firefox profile syncing, and some software to support it, maybe you should check that out (but the actual software FF uses is or used to be FOSS as well).
Brave