Who is doing the assigning of the dynamic ip address, the isp or the router? Because one article I read elsewhere said the ISP, the other said the home router. Can a home router also moonlight as a DHCP (Dynamic Host Configuration Protocol)?
As far as I know, the ISP typically assigns the dynamic IP address, not the home router. Your ISP controls the pool of IP addresses, and when your device connects to the internet, the ISP assigns an IP from that pool. The home router does manage internal IPs for your local network using DHCP, but this is separate from the public IP assigned by the ISP. So, while your router does handle internal IP assignment, it's the ISP that determines your public-facing IP.
You can have your own router IP address pool that is different from the ISP's pool—that's the normal setup. However, you can ask the ISP to bridge your device so you can use the IP address they provide. Going back to OP's concern, if you have a dynamic IP, your IP will change from time to time, but if you have a static IP, that will be permanent and recorded by the casino. However, since casinos are not overly concerned with our IP address, we just need to be mindful of our location if we want our account to remain secure, and avoid changing from one country to another every time we access the casino.