New Post: How to handle unexpected authentication requests properly.

Aye, you don't understand SSH.
SSH has nothing to do with what happens in the shell, it just gives you access to it.
Not putty is hiding the password, the program in the shell hides the password.
The password request is not a SSH password request, it's a password request of the program.
Perforce is asking for a password, not SSH! It happens in the shell, not in the SSH connection.
I think you should read more about SSH, shell and so on.
Without basic knowledge you get lost IMHO.