Normally the Ambari user accounts are stored in the Ambari DB, whereas the user accounts which you use to login via port 4200 are the local unix accounts on the host.
By doing "ambari-admin-password-reset", you have reset the password for the Ambari admin user account. You can utilize this admin username/password only while logging in to the Ambari Web UI via port 8080.
Whereas if you wanted to login to the Web based SSH client in port 4200 as the "admin" user, first could you login as the "root" user and check if the user account is present using below command?
$ id admin
If the above returns "no such user", then please create the "admin" user account, set a password for it and after that you should be able to login successfully as the "admin" user in the Web based SSH client.