Wakanda may be fake, but Wakandan is a real language. In scenes in Captain America: Civil War where actors playing Wakandans speak to each other, they're speaking Xhosa, one of South Africa's official languages.
While T'Challa (Chadwick Boseman) speaks English to the other English-speaking characters in the film, the choice was made that scenes between T'Challa and his father, T'Chaka (John Kani), should be in Wakandan; it is, after all, an island nation with a strong resistance to outsiders. They'd obviously speak in their own language.
On the choice of Xhosa, director Joe Russo told Entertainment Weekly, "John Kani, the actor who plays T'Challa's father in the movie, speaks the language and taught it to Chadwick."
Xhosa is spoken by 7.6 million people, and one of its characteristics is the use of clicks as part of the language. Here's a pre-Daily Show Trevor Noah explaining it:
So when you hear Wakandan in the film, know it's not a made-up language in the vein of Klingon or Dothraki. It's a real one.
Image: Screencap of Black Panther in Captain America: Civil War, Marvel/Disney