I ran into the same issue last week when I tried to reset my Kindle password. It turned out that Amazon requires you to reset the password through their main website, not directly on the Kindle device. You need to go to the Amazon website, log in with your account, and navigate to the 'Login & Security' section. From there, you can change your password. The Kindle syncs with your Amazon account, so once the password is updated on the website, it will automatically apply to your Kindle. If you're still having trouble, make sure you're using the correct email associated with your Amazon account and that your device is connected to the internet.
Another thing to check is whether you've enabled two-factor authentication. If you have, you might need to enter a verification code sent to your phone or email before you can reset the password. Sometimes, clearing the cache on your Kindle or restarting the device can also help resolve minor glitches that might be preventing the password reset.