Gambling sites will use many of the ways listed here to detect the use of VPN. They can also use cookies which tracks your activity when you're not logged into the site and maybe have the VPN off. There will obviously be advanced tools that use to perform these functions.
Can a home router also moonlight as a DHCP (Dynamic Host Configuration Protocol)?
A router can be configured to function as a DHCP relay or proxy but I don't think it assigns UP addresses.