It doesn’t make sense to “protect” against this. If this kind of copying happens, then either:
- The end user did it on purpose because they wanted to change computers. This is, of course, not something you should care about or be concerned about.
- An attacker has already compromised the user’s browser and gotten access to the cookies stored inside. By definition this cookie is a secret that proves that the identity of the HTTP client. If the attacker already has access to it, they can already use it in any number of ways of their choosing that you won’t be able to prevent or distinguish from the real user accessing the server legitimately.