It's not really the same thing, but the question reminds me that we almost had SQLite in the browser[1], but Mozilla spiked it in favor of IndexedDB[2] (yet "Firefox saves Web storage objects in a SQLite file"[3] so I dunno what to conclude from all that. SQLite is good enough for FF devs but not users?)

Anyway, if you have a web app you already have a DB-per-user, FWIW.

[1] https://en.wikipedia.org/wiki/Web_SQL_Database

[2] https://en.wikipedia.org/wiki/Indexed_Database_API

[3] https://en.wikipedia.org/wiki/Web_storage