I have a SharePoint 2013 farm, and I wish to upgrade to 2016 on the same boxes. From my reading, everyone says you have to take the database, and attach it to the new farm. Can I do the upgrade in the same server?
No, you can't do the in place upgrade.only supported way is database upgrade or using 3rd party tools.
When you upgrade from SharePoint Server 2013 with Service Pack 1 (SP1) to SharePoint Server 2016, you must use a database attach upgrade, which means that you upgrade only the content for your environment and not the configuration settings.
Agreed with above suggestion.
Moreover, please refer to this earlier thread having suggested solutions i.e. Step by step upgrade from 2013 to 2016: https://social.technet.microsoft.com/Forums/office/en-US/a31db70c-9d42-46c1-a0ce-6f28a5bfe63b/step-by-step-upgrade-from-2013-to-2016?forum=SP2016
Or you colud try a third party SharePoint migration tool to accomplish this migration task as they can take care of all your data, metadata included.
Hope it helps!