This is a ridiculous issue. For us the workaround was to send the invitation to another e-mail and forward it to the apple id one and claim from there.

The problematic account had been used in another organization as a tester. After removing it from there, I could successfully add it to my organization. This solves it for me, I guess, but the issue is still valid: the error message is misleading it should be possible to add an Apple ID to multiple organizations @robnotyou it is an Apple ID, but that should not be an issue. I'm adding existing Apple ID's without any problems, usually (and I don't think it would be wise from Apple to prohibit that).