the bizarrely shoehorned-in fictional culture of having a “number one” (literally only Picard used that nickname for Riker back in the day)

Number One was "the first character Gene wrote into the script" of The Cage (the pilot episode of the original series), according to Majel Barrett Roddenberry.

There are also many other characters called Number One in Star Trek and elsewhere.

According to some sources, calling the second-in-command/executive/first officer "number one" might have historically been a thing in the British Navy, but i don't see a reliable source for that after a minute of searching so I'm not sure.