Some background on why this doesn’t work:
Packets are routed between networks at the appropriately named internet layer (or layer 3 / network layer in the OSI model). This layer adds the IP header, which contains the source and destination IP addresses of the packet.
The transport layer (or layer 4 in the OSI model) on which UDP and TCP live, adds these protocols’ respective headers, which contain the source and destination ports.
DNS is an application layer protocol (layer 7 in the OSI model).
The domain name cannot be contained in the network and transport layers, there simply is no header field for it. So if you wanted to propagate that information from a client to a server, it would have to be in an application layer header. This is what HTTP does with the Host header, for example. So for a machine to make decisions based on the domain name, it would need to decode the application layer headers. But this requires two things:
a) The machine needs to understand the application layer protocol. For example, reverse proxies understand HTTP and thus can make decisions based on the domain name.
b) On the sender side, the domain name would have to be inserted at the application layer in the first place.
But even if both of these conditions were met, you would still need a “proxy” for every single combination of internet layer protocol, transport layer protocol and application layer protocol on your router. This is simply infeasible.
Btw., these “proxies” do exist on the internet and transport layer. This is what NAT/PAT is. This works well because there is only 1 internet layer protocol (the Internet Protocol / IP) and 2 transport layer protocols (TCP and UDP) which are in widespread use, for a total of 2 possible combinations. Compare this to the literal thousands of application layer protocols.