ISP's that can't be bothered to run their own cluster of IP addresses just use the same trick as every home user and have a range of reserved private IP addresses (10., 172.16., and 192.168. from memory) that they issue to their customers - however the *external* IP can be the same for hundreds or users, or cycled round, or loosely linked to a particular user, etc. just like in the dial-up days.
The decent ISP's now use static addresses (still allocated via DHCP, though, it's just that it recognises your login and gives you the same address each time, assuming everything works as normal) and everyone has their "own" IP, but even some major ISP's in developed countries still squash everybody onto dynamic IP addresses that they choose from a limited pool and/or private ranges. And rebooting / replacing the router doesn't actually change anything about the connection because "MAC" addresses don't exist in most of the telephony-network-side of broadband. My ADSL router has a MAC for my local network, but for the actual ADSL-side of thing, it just uses my login name, password and VCI/VPI pair. It doesn't matter what my router is, that won't change anything. That's what my ISP use to determine which IP address I should get. However, some cable networks do use the MAC to determine the address - however, changing this is frowned upon because it means you've put an "unauthorised" device on the cable system that for all they know might be able to sniff the whole cable for the street (this is why lots of cable routers let you "fake" the MAC of an authorised box that you already own).
Most BT-based ADSL ISP's, and some of the cable companies, in the UK offer static IP's but not all, by a long shot. And even then, sometimes I see my IP switch to an "emergency" BT IP and I just have to wait for things to be fixed, no matter how much I reboot / change the router.
That said, who cares? If the IP is banned, it's banned. Using a proxy / VPN will "change" the IP, but it's more important to find out how/why the IP got banned in the first place. It suggests misconfiguration, accidental banning of the one of the forum's most important members, over-zealous anti-spam scripts and / or a deliberate change by a malicious user.