I want to play an online game with my brother, but the game seems to prevent having two users use the same external IP Address at the same time. Is there any way to have two different external IP Address for my computer and my brother's, besides using proxy/VPN (because it becomes really slow)?
On the one hand, yes, it's possible; it's how routers work in the first place – by routing IP packets between computers with different addresses.
But on the other hand, it's not going to work, because most likely your ISP only assigns you one global (aka external) IPv4 address. (Partly because IPv4 addresses are in very short supply – the internet is only slowly moving to IPv6 – and partly because ISPs like to charge extra for things like extra IP addresses.)
(The reason you can connect multiple computers to your router/gateway is because the router has a NAT function ("address translation"). With NAT enabled, it assigns "local" (or internal) addresses to all connected computers, and the "global" ISP-assigned address doesn't belong to any of your computers – it belongs to the router. The NAT function rewrites every IP packet to make it look as if all computers are using the same "global" address rather than their own local ones. This is also the reason many games and P2P programs ask you to set up "port forwardings".)
If you call your ISP and ask them for a second IP address to be added on your connection – which usually costs a bit extra – then it's of course possible to reserve a specific global address for a specific computer. (Though there's the question of how configurable your router is; some devices can achieve this in several different ways, and some other "home gateways" have just the bare minimum of settings.)
The only other way beyond a proxy would be to either have 2 separate internet connections at that location or get an additional IP assigned to the existing connection and then setup the firewall/router to NAT your traffic out one IP and his out the other IP. But consumer/home ISP accounts vary as far as allowing you to buy additional IP addresses (whereas with a business account it is almost always an option).
A hack alternative would be to use a 4G connection for him, either tethered via phone or 4G hotspot. But bandwidth and data consumption might be an issue there.