Because the songs were written by the superstar team that wrote the songs for the movie: Alan Menken and Glenn Slater.
Alan Menken is also responsible for the songs from such Disney movies as The Little Mermaid, Beauty and the Best, Newsies, Aladdin, Pocahontas, The Hunchback of Notre Dame, Hercules, Enchanted, and Ralph Breaks the Internet, as well as many, many other things, Disney and not-Disney.
Glenn Slater has worked with Alan on several projects, like Home On the Range, Galavant, Ralph Breaks the Internet, and The Little Mermaid Live!
Not to mention, they have professional singers contributing their voices to most of the songs (definitely not all of them, though).
Because of their close collaboration on so many projects, not only do they understand each other, they also understand the range and skills of many of the singers they were writing songs for. When Glenn answered questions about Tangled songs for the movie's 10th anniversary, he mentioned that knowing Zachary Levi's voice made it easy to write songs for him to sing. Similarly, Alan has worked with Jeremy Jordan (Newsies: the Musical) and James Monroe Iglehart (Aladdin: the Musical) before. I don't know if either of them had ever worked specifically with Eden Espinosa before Tangled: the Series, but they clearly understood her range and voice type, too, since "Waiting In the Wings" was the only song from Tangled: the Series to win an Emmy.
I can't speak for other Disney shows, because there's a lot I haven't seen, though I do know they sometimes get top performers (like Mandy, Eden and James, who have all been in more than one Disney series).
But the series set out to have amazing songs, right from the get-go. That was one of the things they definitely wanted to accomplish when making the series, which is why they got the movie's composers and lyricist back for the series.