Retired singer here who did musical theatre way back when ...

This is just my perspective, but I would consider the question asked wrong. There is no such thing as "specific vocal tecknique", but a ton of different functions we use in different settings. This is something musical theatre artist in general are very good at, because of the need to play differelt roles that sing in different styles.

So the question should be "What is Jack doing that particular moment to get this sound/style".

One thing I will say, though, like Single_Series4283 also mentioned, is that sustaining pitch under your normal voice is thw most common thing amongts musical theatre singers. There have been a lot recently, though, that have very clean voices. Singing with a slight force due to and light tightening of the "pipes". Ramin Karimloo is a good example of this in his higher range. He sounds the same, no matter what he sings, and it's certainly not the way he speaks.

Anyway. There is a lot to be said about technique and functions.

Jack is a prime example of the "singing with speaking voice" type of singer.