One possible solution idea I just had is having the option of registered providers (such as Discord). They would have a public key, and the user has a private key associated to their eID. They could be mingled in such a way to create a unique identifier, which would be stored by the provider (and ofc the scheme would be such that the provider can verify that the mingled identifier, was created from a valid private key and their public key).

This would in total make sure that only one account can be created with the private key, while exposing no information about the private key aka user to the provider. I am fairly certain that should work with our cryptographic tools. It would ofc put the trust on the user not to share their eID private key, but that is needed anyway. Either you manage it or it gets managed (and you lose some degree of privacy).