This seems really bad to me. There are so many ways for a website to end up in one of my browser tabs without me wanting it there, or even knowing it's there.
If that happens, and that tab just so happens to be a malicious MCP-B enabled page, it could steal all kinds of data from all kinds of different web apps I'm interacting with. I think it should be seen as the responsibility of the framework to enforce some level of data isolation, or at the least opt-in consent mechanisms.