Yes, this is a real threat .
"Remember me". A cookie puts the security of your web service out of your control, by definition. Now, in general, anyone (especially a sophisticated attacker) that can capture this cookie can log in as this user.
Let's look at an example of the real world: Google uses these cookies for its services. You can log in in a few weeks. From what I have observed, a way to mitigate attacks using cookie theft is to invalidate the cookie if they detect suspicious activity on the server side. For example, if I usually logged in from California and unexpectedly logged in from another state / country (or you have concurrent sessions from another location!), I might have logged out and had to re-authenticate. Of course, this is not perfect, but usage patterns can be used to prevent some attacks.
Also, remember that the cookie will be browser specific. For example, if a browser fingerprint was used to determine that a user had just logged in from another OS / browser / etc., this might be a good time to invalidate the cookie. Perhaps you can get some imagination and allow some freedom if the small version of the browser is updated, but check it if the version of the browser is ever downgraded.
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