Well. I'm learning as I go and I haven't seen the series. But here's what that sounds like to me.
You've identified rhythm and meter. Maybe these can be imagined as dialog that varies in its staccato or legato elements.
Here are a few more ideas.
1. There are different genres of music. (and different types of dialog can work.) One type of music is the blues. Here are some lyrics from Muddy Waters:
Baby please don't go.
Baby please don't go.
Baby please don't go, down to New Orleans.
You know I love you so.
The pattern in the blues is repeat, repeat, change.
In a lot of other styles of songs (and maybe blues too), there is a pattern of verse, verse, bridge, verse.
So, music, even music that is geared to adults, has elements of both repetition and novelty woven into it.
Dialog can do this sometimes too. The repetition can have the effect of reinforcing certain ideas. The repeats can be spaced or immediate.
'What are you doing?'
'What am I doing? I'm leaving.'
This ain't a great example, but I think the idea of repetition is worthwhile.
2. Music has chords, multiple notes, multiple instruments, and also high notes and low notes. Put that complexity into dialog.
Pairing characters with opposing values in dialog can move a story. Or simply be interesting. Imagine a mother who wants to protect and raise her child safely, and a child who just wants to go out and see the world. What is their dialog?
Or an angry boss and a meek employee. Or a single character arcing from patience to frustration during a conversation.
The dialog can reflect these 'opposing' pairs, as 'high and low' notes.
3. Music has harmony and balance.
Dialog should, too. If you have two characters arguing a point, they will not each state their case in completion and then listen to the other case in its completion. Instead they will go back and forth, and the amount of this from either character should balance out.
4. Chopsticks is hard to listen to.
So is dialog that repeats a sound again and again.
'Stop shaking the salt shaker.' is hard to read. 'Will you stop that? You have plenty of salt.' is easier.
5. Playing by ear.
I think Sorkin's right? It seems like a person picks it up with time. I don't know if it can be taught or not but I think it can be learned.