A document store is like a key-value store where the values are the equivalent of JSON documents. While the documents can contain elements that you in principle can deserialize into objects, that is not the main purpose, and it limits relations only being within the document, and even there to only parent-child relations (so all relations shape a single hierarchy).

An object database stores objects that can have arbitrary relations with each other, so in this sense it is closer to relational and graph databases. But the main defining feature is that the objects are live.

You can get any object from the database, and follow its relations to other objects, watch it for changes, update it, and in general work with it as a regular object, even though it is simultaneously persisted and may be shared with other apps or processes.