Android keeps asking "Sign-in to WiFi network" even if WPA/WPA2 is already authenticated
It's a problem with your router. After Android successfully connects to a Wi-Fi network, it tests to see if it can use this network to access the Internet. This is because it's commonplace for Wi-Fi networks to redirect all requests to some kind of login page, until you've logged in. The "Sign-in to Wi-Fi network" notification is nothing to do with authenticating to the network itself: it appears when Android detects that the network is redirecting requests.

Given that you've said any page brings up this 404 error page from your router, that fits perfectly with the fact you're getting the notification. The fact that the router won't give you pages from the Internet is the root problem: it's nothing to do with WPA or anything like that. Does the 404 page tell you anything helpful?

Perhaps the router is set up with a list of devices (by MAC address), and the new device isn't on the list so the router won't give it Internet access.