That's application specific. In firefox you can set a main password to protect your saved passwords.
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https://support.mozilla.org/en-US/kb/using-master-password-firefox-android
The reason Linux asks for your password to install programs is to prevent users without the password from installing and to give you a 2nd chance to ponder if you really want to install HaXXor Pwned.exe.
Nothing to really fix because thats the design. In userland you can do anything, system wide needs elevated credentials. If you don't want the password prompt, look into aliases and sudoers/pkexec
Sounds like a browser config issue rather than a linux issue
Probably heavy depends on your process to save and use saved passwords. I personally stopped using the web browsers to save my passwords.
It asks for password on brave browser. Seems like a browser setting issue.
Edit: I use latest Linux mint.
This depends on browser support. I recommend bypassing the issue altogether by using a password manager like Bitwarden (a free as in freedom SaaS) or KeepassXC (a program for using .kdbx password databases which can be synced between devices by you in any way you want, like Nextcloud or Dropbox).
You can configure sudo so that it doesn't require you to enter a password. However I would strongly discourage this, since it also allows any script or program to execute root commands without your approval.