We are doing our first cumulative update on our SharePoint 2013 on-premise. I know that when doing an update the databases are also updated/modified. But in the interest of backups and storage, would anyone be able to tell me which databases are affected, or is it all of them? We are two years behind on updating.
Not all databases are updated with every patch (although a CU contains all previous patches, so they certainly may be).
For backups, regardless if a database schema is updated or not, you should be backing them all up in order to perform a restore as a monolithic entity, with a few exceptions. You can skip the Usage logging, State Service, as well as User Profile Sync database. Technically you can also skip the Configuration and Administration databases since those aren't supported for restore.