If your router or scanner is to be accessible over the network then it needs its own name and it needs to be able to certify that that's its name. Anything else is just too dangerous. A user expects addresses they enter into the browser to mean the same thing on any connection; having a few "magic" addresses that go one place on one network and another place on another network is a recipe for users getting hacked.

For the consumer use case, maybe the router gets a unique default address in the manufacturer's namespace (router12345.linksys.com) and ships with a certificate for that name and that name printed on the box, just like we do for the admin password. Since it's a router it's probably running the DNS for your network (at least in the consumer use case) so it can route requests for itself correctly. For scanners or similar, the router would need to update its DNS when the scanner joins the router's network - a lot of routers already do this within the local domain based on DHCP registrations, so this ought to be simple if it's not already done. Crucially this part isn't security-critical - if you try to print a confidential document on your network printer while you're on your neighbour's wifi, the worst their router can do is not route you, because an evil endpoint won't have your printer's certificate.