@Ken Fly Thank you for reaching out to Microsoft Q&A, apologize for any inconvenience caused on this. yes all will be using the same
Connecting to SSH for Linux containers using either Web ssh method or Remote-shell method requires authentication, and you need to pass your azure subscription credentials.
Remote-shell method would have worked for you since you might have tried to SSH into Linux container of the app service through cloud shell from portal or command line editor from your local machine where you have already been authenticated already. In other words, refresh token is not expired from the last session.
Expected behavior is that the user will be prompted irrespective of the SSH method provided the refresh token is expired.
Feel free to reach back to me if you have any further questions on this.