Changing password actually depends on the type of wallet you are using, some wallet require some verification via sms or Gmail before you are being able to change your password and no one will be interested in wasting time doing all that.
It does not depend on the type of wallet if it is a noncustodial wallet. Only centralized platforms can do that and they are completely different from noncustodial wallets.
Yes platforms like centralized exchanges (binance and its kind) send new passwords to your Gmail and also phone number used to register on them. This one of the downside of using centralized platforms because this password are gotten with your email it is much easier to get access to your exchange.
After generating the same password, will the new wallet have the same address as the original wallet?
Let me break it down for you. A password only works on application or wallet you set it. For example if you set a password on the MetaMask wallet, when you import the seed or keys gotten from that wallet to a new wallet (say blue wallet), you do not need the password set on MetaMask to access this funds because the password only applies to the MetaMask wallet on that device only