Why is it possible to use the same port on TCP and UDP at the same time?

The destination isn't identified by IP Addr:Port alone. There is another thing - IP header has a field called Protocol which differentiates the TCP and UDP endpoint. As such it becomes possible for two process to bind to same IP:Port as long as communication protocol is different.

The other answers are correct but somewhat incomplete.

An IP (aka "INET") socket "connection" (i.e. communication between two processes, possibly on different machines) is defined by a 5-tuple: protocol, source address, source port, destination address, destination port. You can see that this is not limited to a stateful connection such as TCP.

This means that you can bind different processes to any unique instance of that 5-tuple. Because the "protocol" (e.g. TCP and UDP) is part of the differentiating factor, each can have a different process.

Theoretically, you could bind different services to the same TCP port if they bind to different interfaces (network cards, loopback, etc.) though I've never tried it.

It is standard practice, however, to always use the same service on the same port number. If both UDP and TCP are supported, they're just different ways of communicating with that same service. DNS, for example, uses UDP on port 53 for lookup because they are small requests and it's faster than creating a TCP connection but DNS also uses TCP on port 53 for "transfers" which are infrequent and can have large amounts of data.

Lastly, in complete accuracy, it isn't necessarily a 5-tuple. IP uses the "protocol" to pass to the next layer such as TCP and UDP though there are others. TCP and UDP each seperately differentiate connections based on the remaining 4 items. It's possible to create other protocols on top of IP that use completely different (perhaps port-less) differentiation mechanisms.

And then there are different socket "domains", such as the "unix" socket domain, which is completely distinct from "inet" and uses the filesystem for addressing.