In short: Game and streaming sessions last hours and break when the address underneath them changes, so they need an address that holds. A static ISP address keeps one IP for the whole paid period.
What the protocol and the network allow
SOCKS5, specified in RFC 1928, relays TCP connections and also defines a UDP relay. Real-time game traffic often travels over UDP, and support for that part varies between implementations, so test your client rather than assume. Mobile addresses fit badly for a different reason: carriers commonly share public IPv4 addresses among many subscribers through carrier-grade NAT, whose requirements RFC 6888 sets out, and a carrier can hand a device a different address when its data session reconnects. What this page says follows from the protocols and from how each kind of address is assigned. We have not measured latency or session length on any game or streaming service, which is why the procedure below has you test your own.