Do Portal folders have to be on the Desktop?

No! You can create your folders for your Portals in whatever location you like. (Or use existing folders).

For example, you could create the following folder:
C:\Users\<username>\Documents\PortalFolders

Then in that folder you could create folders e.g.
Games
Applications
etc.

Then create Portals from those folders – this way your desktop remains clean!