If I provide you with a SHA256-hash of my name, would it be in principle possible for you to verify whether the hash is indeed the output of my name?

In theory, yes it would be possible to verify this. However, in case I don’t know your name, I would need to enter a name in a SHA256-hash generator, and by trial and error check whether the hash output of my input is the same as the given hash. In practice, this would be almost impossible, but in case I’d know your name it would be done quickly.