What embedded database format is used by this Chrome extension?

It’s LevelDB, a key-value store.

You can use NodeJS and levelup to access the data:

var levelup = require("levelup");
var db = levelup("path/to/directory");
db.createReadStream().on("data", data => console.log(data.key, " => ", data.value));

(levelup requires leveldown to be installed for accessing on-disk databases.)

The files are in LevelDB format developed by Google and the code is hosted on GitHub.

Potentially you can use leveldb-json utility to export the data by pointing to the .indexeddb.leveldb folder, however, you would need to implement a compatible comparator first (as Chrome provides its own comparator implementation) in order to inspect Chrome's Indexed DB leveldb instances as per information at How to access Google Chrome's IndexedDB/LevelDB files?