The name of your DEVONthink databases doesn’t really matter. While it matters to you in terms of helping you to identify what is in it, the real identity of the database is found in its UUID (universally unique identifier). You can think of it like your social security number (in the US) or your passport number. You could change your name, appearance, location, etc. but this number always represents me.
When a database is created, a UUID is automatically generated for it. This value, a long alphanumeric string, is a constant that is used by all underlying mechanisms, including the synchronization. Say, you have a database named ABC and it has a UUID of 123456. If I sync it, then the sync location stores its data in a location referencing 123456. If you decide to rename the database to MyStuff and sync it, it’s still syncing database no. 123456, not a new database named MyStuff.
This is also why you should not copy a database and simply rename the copied file. You are copying the UUID of the original database, regardless of what you name the copy. You will end up with conflicts and potentially data loss.