Most sites will have a confirmation once you click the link that includes the browser version and IP address. I have seen that info only in the email itself too with no confirmation afterwords, but not for some time. Have never seen one that is just a link with nothing else that once clicked allows the other device in but supposes could be implemented that way.
The article itself is about not making them the only option (which is fair), and the OP says if they do it should login the device which originally made the request (which I agree). If the implementation is just an email with only a link, no other information with no confirmation (yes, it's fine to let this device in), then I would have to agree with you it's very risky and could allow anyone to login as you (hopefully no sites are doing this, but...)
If you really want to allow for another browser to authenticate a login request, you can at least limit it to sessions coming from the same IP.
That would let you authenticate your desktop browser from an email you opened on your phone if you're on your home network, but without becoming widely exploitable by phishers.
Some people will still click the button because they expect it will give them more information about why they received the link. You can add text along the lines of “authorize login on $other_device”, but it’s still risky.
