Arduino Disconnects after router password changes
Routers have multiple "passwords".  I assume you are talking about your wireless encryption key (i.e. WPA/WPA2 key).  When the encryption key is changed, you are correct: you have to re-program the ESP8266 with the new key, because it can't learn this information on its own.  If this key were broadcast, then your network would not be secure as everyone would be informed of the encryption key.

Why is the key changing?  Is there really a need to change the encryption key often?

MAC addresses can be spoofed, so permitting a specific MAC address to "always connect" is not secure.