You’ll notice that on the official IANA list of port numbers and service names, there is a separate section, after the numbered ports, listing only service names¹. Just apply for one of those.
1. At the time of writing, starting on page 135 of the HTML version: <https://www.iana.org/assignments/service-names-port-numbers/...>
But getting a port assignment these days is going to be virtually impossible. It’s probably not going to happen without at least a Proposed Standard RFC.
https://www.iana.org/assignments/service-names-port-numbers/...
There’s not a single RFC published that concerns redis. It’s not going to happen.
I needed to write a justification why the organisation I represented needs a port number (a custom binary protocol), a formal confirmation that protocol has versioning built-in (so we will not request for a new port number for the next version of the protocol), a confirmation that we have a running code implementing the protocol (if I recall correctly, link to documentation was sufficient), and the reasons why we cannot use any of the existing protocols.
It’s not impossible and I think it’s totally doable for redis unless the port is already reserved for something else.
If there was a moderately complicated bureaucratic process to be assured of getting one, it wouldn't be Redis or the next SSH that got assigned ports. It would be Oracle or HP or someone else incapable of pretending to be a good Internet citizen, who filled in 10,000 of the forms promising that they have 10,000 totally necessary well known enterprise services.