Quick Answer: The refusal to say "zombie" stems from a film and television trope known as "genre blindness," which refers to the way characters in a particular genre story are unfamiliar with the commonalities of the genre. Just as horror movie characters consistently repeat the actions that get them killed, people in zombie stories have never heard of zombies.
The Walking Dead (2010 - ) is a show about people navigating the challenges of surviving in an apocalyptic world overrun by zombies. They kill zombies, they defend against zombies, and they try not to become zombies -- but you’ll never hear the word “zombie” spoken on the show.
The creators and writers of The Walking Dead are vigilant about ensuring the word “zombie” is never said in-world. This isn’t because the word is trademarked or they are avoiding the acknowledgment of their reality. Rather, it's the result of a narrative trope called “genre blindness.” Genre blindness is the reason plots work in narratives that include extreme circumstances. Why are teenagers always running away from killers in slasher films, or traveling solo into dark creepy buildings in horror movies? Because they’ve never seen a horror movie and don’t know any better -- that’s genre blindness.
In zombie stories, people have never heard of zombies. That’s why the characters are so unprepared. That’s why it takes time before they realize they need to attack the brain and not the limbs in order to take one of these monsters down. Almost always, people fighting zombies in fiction have never heard of them before, as they are blind to the genre they are within.
Some fine-looking walkers.
The term zombie is actually an old Haitian word referring to people who are being controlled by voodoo. Early zombie stories like Victor Hugo Halperin’s 1932 film White Zombie are based on this premise of losing one’s consciousness to voodoo mind control. It wasn’t until after George Romero directed the seminal horror flick Night of the Living Dead in 1968 that zombie became a term for the reanimated dead. And even then, it wasn’t actually said in films. It’s a retrospective pop culture term ascribed by the masses in reference to these films.
In Shaun of the Dead (2004), Simon Pegg and Nick Frost poke fun at the popular concept of genre blindness and people not saying the word zombie in zombie films. Here, Pegg’s character chastises Frost for using the word.
On Talking Dead (2011 - ), the Walking Dead after-show, creator Robert Kirman said, “One of the things about this world is that people don't know how to shoot people in the head at first, and they're not familiar with zombies, per se. This isn't a world the (George) Romero movies exist in, for instance … because we don't want to portray it that way, we felt like having them be saying 'zombie' all the time would harken back to all of the zombie films which we, in the real world, know about. By calling them something different, we're kind of giving a nod to … these people don't understand the situation. They've never seen this in pop culture, this is a completely new thing for them."
In other words, “zombie” is a colloquial concept word we’ve created for fiction. In The Walking Dead, this fiction is manifested as reality without prior reference. In that universe, the fiction doesn’t exist, so they have to come up with their own terms for what is happening around them.
The most popular term for zombies on The Walking Dead is the one utilized by the series’ central cast: walker. For them, it refers to the animated dead’s most basic level of ability as ambling corpses.
Each time the group encounters other people, we learn of that group’s particular terminology for the dead. The Governor’s (David Morrissey) people called them “biters,” and The Governor himself often used “lurkers.” ‘The Termites preferred “cold bodies.” The Alexandrians typically go with “roamers.” We’ve also heard “lamebrains,” “geeks,” “floaters,” “creepers,” and “rotters,” among other terms, throughout the series’ run.
We will likely continue to see the genre blindness trope in effect. Without it, many fantastic stories wouldn't work. Too many narratives rely on their characters' ignorance of certain realities to create a foundation, and genre blindness is a means to that end.