ArcGIS Pro requests the user credentials each time while connecting to an existing database connection in a saved project, even if the Save Username/Password option is checked after updating the password of the database user.
This issue is non-reproducible with the current software version. During the ArcGIS Pro 3.4 + development cycle, effort was made to fix several issues as part of the work to build a newer interface for the database connection dialog. If the issue persists with the current release, please contact technical support.