Could you suggest if there is any reason to compare cookie value domain, path, etc with web "location.href" and so on? I mean - on the client side.
No. I don't know of any reason to perform that sort of check in a browser plug-in. Web browsers will already perform same-origin checks for you. But then, your question was pretty vague, so it's hard to know what exactly you might have in mind.
Generally speaking, I prefer to start by asking: What problem am I trying to solve? Once you know what problem you're trying to solve, then you can start to talk about technical measures that might help.