There are hotspot routers. e.g. this one:
New Router Can Turn a Small Business into a HotSpot - www.smallbusinesscomputing.com
But a simple free hotspot would likely be a wireless router that's firewalled and another that's for employee use.
Not sure how they do the weird things:
1. Log in page with a 1 hr pass or how
say t-moble has the login page as well.
At hoe Verizon sometimes lets their search page in edgewise somehow.
Bandwidth determines how many people can be on with reasonable speeds.
The wifi hotspots would likely use an access point for wireless.
The "router" could be a Linux box.
We had a weird system and I don't know how they did it. We had fixed addresses on a subnet(A). Then we had DHCP addresses on subnet (A). If the mac address was "registered" with the system, you would get subnet (A) otherwise you would get an entirely different private network where basically what you could do is register a machine.
Conjecturing: registered MAC address return legal IP addr
Unregistered mac address return something on say 10.0.1.x.
The gateway could be a port on a machine. Once you registered, eventually your IP addr would change to the real one. It probably has a short lease.
Never could figure it out.