So we all agree that this feature would be great to have, with just the one drawback that it is impossible to build, as proven by the fact that even LastPass can only make it work for a couple of dozen sites.
How about we build a solution that does not involve the impossibly difficult task of interacting directly with change password pages. I suggest a solution that gives the user a simple interface containing the old and new password, and the ability to copy/paste the old and new password into the change password page of any given site, and update the password in the database when done. It’s totally doable, it will work for all websites, and it will not be hard for users to understand.
I wrote up a fairly detailed proposal for this a while ago, linking here: