I'm using SqlServer and the application role for my application, but I also have username in SqlServer.
Is there any possibility to change user password when using the application role?
When I'm trying to change user password I get an error, that I don't have permission to change it.
I granted an alter permission to the application role, but it didn't help.
AFAIK, this is not possible and really isn't the use case application role was designed for.
Application roles are database level objects, or more accurately principals since it is a security object. Logins are server level principals. You cannot grant a server level privilege to a database level principal.
It really isn't the use case for app roles but if you absolutely must do it, you can create a stored procedure in the database used by the app and configure it to execute under a different principal but it's hacky and I wouldn't do it for a production system.