My SQL Server 2014 Express database has reached database size limit.
Correct solution would be to upgrade it to version which does not have limitation on data file like developer, Standard and Enterprise depending on your requirement. If there is space in your database try shrinking it and see if that helps. But this is temporary solution in no matter of time you would again face space issue.
Is Express software the same as the full version, with just some limitations, or is it a complete different software?
No its is not same as full version there are many differences and all is documented in Features Supported by the Editions of SQL Server 2014.
Can I just go to Microsoft store online, order a key for full version, and type it in to the current Express to activate full features, or do I need to do full uninstall of Express, reinstall of full version, copy database and such?
You can perform an in place upgrade,you don't need to uninstall the express edition. Please see Supported Version and Editon Upgrade. You need license key mostly its embedded in new installation setup you get.