Earlier quoted context omitted.
So are you bridged then or is it really a double nat?
this is really where my limited knowledge of networking shows. I'm not entirely sure but I want to say both or double nat. There's two networks, but my router thinks theirs is a modem and is connected via the "Internet" port, not just a normal device port
Edit: to go into more detail, their router is acting as a NAT for your public IP, giving you your first subnet, and then your router is getting a single IP on that subnet and creating a NAT where your devices all get IPs. In a bridge there would only be 1 IP space behind a single NAT. In your case with a double NAT a lot of consumer things might not work (like UPnP) and port forwarding would require you to add rules to both routers.