This is a known issue and as of April 2025 we are working on it. Mysql 8.4 introduced a new password strategy called "caching_sha2_password". Cloud Run uses the Cloud SQL Auth Proxy to the Cloud SQL database. There is a bug in the Auth Proxy (and other connectors too) that breaks the caching_sha2_password protocol. The login starts working again after you log in with Cloud SQL Studio because the authentication is cached in Cloud SQL instance for a period of time.

We are tracking the bug here and actively working on a fix. See Cloud SQL Proxy #2317

At the moment, your best work-around is to downgrade to Mysql 8.0.