One of the best things about This Is Us is the quiet pop culture references that help ground the Pearsons' world in ours. They anchor the different eras, like Kevin and Sophie going on a date to see Good Will Hunting in 1997. They also tell viewers a bit about the characters' tastes, like Jack and Rebecca watching The Shining. And there's nothing better than knowing your favorite character loves the same music or shows you do. That's why it was such a joy when Beth referenced Grey’s Anatomy in This Is Us.
In Season 4, Episode 15, "Clouds," Randall took his first crack at therapy after Beth pushed him to go. It didn't go well. Randall was too defensive, too on edge, and too distrustful. But when he came home and declared it a failure right out of the gate, Beth told him he couldn't fail and that she needs him to go to therapy, for her sake as well as his. Without it, Beth said she can't talk to him about her problems, because he's too close to cracking. She needs to be able to talk to him: "You're my person," she told him, quoting the famous Grey's Anatomy line.
Grey's Anatomy may no longer be the buzziest show on primetime, but the series is a touchstone of 21st-century television. The medical drama started airing in March 2005, and a decade and a half later, it's still going. (Season 17 is already confirmed to arrive in September 2020.) For the characters on This Is Us not to be aware of it would be unthinkable, the same way they've almost certainly watched an episode of The Simpsons (airing since 1989) or Law & Order: Special Victims Unit (airing since 1999).
When Grey's Anatomy debuted in 2005, both Kate and Beth would have been the show's target demographic: unmarried women in their mid-20s, trying to find their place in the world, and, more importantly, finding their "person."
As fans saw in Season 3, this was also the era when Randall proposed to Beth (a lot), in hopes of finally getting a yes. It's easy to imagine he would have eagerly sat down with her to watch the growing block of Shondaland programming on Thursdays, both before and after they got married. For so many IRL couples with similar experiences, lines like "you're my person" aren't just fun references — they're a shared experience.
That's why Beth dropping it during their conversation was so meaningful. She wasn't just telling him he's her partner; in three words, she reminded him of an entire history they share. And it worked, since Randall ended up going back to therapy and trying again.