Is it possible that the next great American media mogul is a woman who would rather sit in bed watching hours of “Grey’s Anatomy” than sit courtside at a Los Angeles Lakers game?

Is it possible that she has made crowds of thousands of women scream over her sex tips?

Is it possible that, on her podcast, she has invited rappers to smoke weed and actors to share trauma, but dismissed overtures from the White House?

Alex Cooper is making an argument for all of it.

Ms. Cooper is best known as the host of “Call Her Daddy,” a podcast that took off in 2018 as the young woman’s answer to Howard Stern. The podcast encountered some well-documented turbulence when Ms. Cooper split with her co-host in 2020, only to land a $60 million three-year deal with Spotify in 2021. As the show aged, it became less bawdy and outrageous, more confessional and sincere.

What is less known is how, over the last year, Ms. Cooper has been building a media company with Matt Kaplan, a film producer and her fiancé.

The company they founded and funded, announced in June as a “Gen Z media venture,” is called Trending. It forms an umbrella over Mr. Kaplan’s Ace Entertainment — he produced the Netflix teen romance franchise “To All the Boys I’ve Loved Before,” among others — and Ms. Cooper’s Unwell network, which currently oversees “Call Her Daddy” and two other podcasts.