IndexedDB is a standard and can be implemented however the user-agent sees fit. Chromium source tree has an implementation on LevelDB and an implementation on SQLite; I'm not sure how they pick the appropriate backend. Firefox and WebKit both appear to use SQLite as the backend.

WebSQL was a clunky API, but not as clunky as IndexedDB which is truly yucky and very easy to get wrong in modern apps that use promises.

I think chromium still uses leveldb, but have been working towards changing to sqlite for a while. https://issues.chromium.org/issues/40253999