Where does google get this data from and is it possible to manually obtain this data on my own website without it (assuming it is in some kind of cookie/user-agent string/identifier/etc)?
Google gets its data from the 3rd Party double click cookie.
Here is a bit more detail from the Google Analytics documentation
If you are looking to get similar data - you can purchase it from services like Comscore but you would need to get something to identify your users against their database.
Why don't you want to use cookie/ user-agent? My personal perspective is that even if users block cookies- then some data is better than no data.