You’re running into an ipv4-vs-ipv6 issue; the Pi is connecting over IPv6, and either the IPv6 detection isn’t working correctly when you try to claim from your browser, or the public v6 addresses of your Pi and your browser are sufficiently different they’re not considered on the same network.
Try visiting https://flightaware.com/adsb/piaware/claim/98395c99-xxxxxxxxx16e with your full feeder ID; this should skip the IP-based path and just directly claim that one feeder. (If you’re using a piaware sdcard image, this is where the “claim this feeder” links on skyaware / the top-level status page go to)