Handmade postcards are officially allowed on Poscrossing, as described in the FAQs:
“What can I send on Postcrossing?
Postcrossing is a postcard exchange community, so each exchange that you make must include at least one postcard. It can be a postcard you bought in a store or a handmade postcard, as long as it hasn’t been previously used.”
However, this doesn’t mean anything with a stamp and an address label will be considered a postcard! If in doubt, contact the Postcrossing team and send photos of your received item.
A few things I received I didn’t need to register.