They call him "the other guy", "the guy in blue", "my cousin", "him", "the male", etc. but they are clearly avoiding referring to him as "Superman". Is it a production choice to keep focus on this show? Is it a legal issue? (The answer must be out of universe because it's clear from the show he is known by that name, it's just not said.)
Out-of-universe, it's likely just a technique used to prevent viewers from dwelling on the humongous 'elephant in the room' (e.g. where's Superman and why aren't we watching him instead?).
They do actually reference him by all of his comic-book names; Superman, The Man of Steel and Kal-el so it's not like he was totally excluded, just not name-dropped eight times.
When I arrived, I was still a 13-year-old girl. But in that same time my cousin Kal-El had grown up and revealed himself to your world... as Superman. The most powerful man in the universe.
The big question: is there a connection between this mysterious flying woman and the Man of Steel?
The show's Executive Producer addressed the lack of Superman in an interview with CinemaBlend:
I think the reason why Superman’s not in the show is because it’s not about him. And it’s her show and it’s about her and he is just sort of a shadow in the back of her psyche, and I think that’s the right way to portray him because that’s how she experiences it.
In-Universe, I suspect that people avoid saying his name for a variety of reasons. Jimmy, Kara and her sister Alex have all trained themselves to be discreet (since you never know who's listening) and her boss; Cat Grant seems to be somewhat more scathing and ironic about him, basically refusing to call him by his given name.
Around the 2:20 minute mark:
"When I arrived I was still a 13 year old girl. But in that same time my cousin, Kal-El, had grown up and revealed himself to your world... as Superman. The most powerful man in the Universe."
That's the first of at least three mentions of "Superman" (and Kal-El is said even from the first few seconds of the episode).