GDPR recital 64 says:
The controller should use all reasonable measures to verify the identity of a data subject who requests access, in particular in the context of online services and online identifiers. A controller should not retain personal data for the sole purpose of being able to react to potential requests.
The GDPR does not specify exactly what methods of verification are "reasonable".
GDPR Article 11 paragraph 2 provides:
Where, in cases referred to in paragraph 1 of this Article, the controller is able to demonstrate that it is not in a position to identify the data subject, the controller shall inform the data subject accordingly, if possible. In such cases, Articles 15 to 20 shall not apply except where the data subject, for the purpose of exercising his or her rights under those articles, provides additional information enabling his or her identification.
This is amplified by Recital 57 which says:
If the personal data processed by a controller do not permit the controller to identify a natural person, the data controller should not be obliged to acquire additional information in order to identify the data subject for the sole purpose of complying with any provision of this Regulation. However, the controller should not refuse to take additional information provided by the data subject in order to support the exercise of his or her rights. Identification should include the digital identification of a data subject, for example through authentication mechanism such as the same credentials, used by the data subject to log-in to the on-line service offered by the data controller.
GDPR Article 12 paragraph 6 provides:
Without prejudice to Article 11, where the controller has reasonable doubts concerning the identity of the natural person making the request referred to in Articles 15 to 21, the controller may request the provision of additional information necessary to confirm the identity of the data subject.
All of the above seems to indicate that the Data Controller must follow reasonable practices, but is not required to guarantee a positive response to a Data Subject who cannot supply reasonable evidence of identity. Moreover the Controller is required to reliably ascertain the identity of the Subject before providing information or acting on a request.
Incidentally, good security practice is that the site does not retain the actual password. instead it hashes the password with a good cryptographically secure hash function, and stores the hash. This means that the site cannot reliably determine if a password is unique among all passwords saved by the site, and searches among passwords are not normally done, and would not be reasonable.
A site using an email address as the User Identifier would normally enforce at the time an account is created that the email is unique among all emails registered on the site. A user who cannot provide the email would typically be rejected as not able to be identified, unless the site retains additional ID info not listed in the question. I think such a practice would be considered to be reasonable in the current state of technology.