I wonder if this process would work if the water were replaced with milk. Milk would degrade at high temperature, but this (at least in bulk) is at room temperature.
I'm surprised you associated it with espresso. Creamer and drip coffee machines go together. Milk and espresso machines go together. I've seem drip coffee machines with real milk, but I've never seen an espresso machine with just creamer.
It tends to exist primarily in office environments, because it can stand not to be refrigerated and was the first lactose-free option. Therefore it was easy to buy one bottle and solve the problem of "stuff to add to coffee".
I make espresso at home by grinding/tamping it myself so I admittedly don't know much about the pod version or whether that still counts as espresso.
It's marketed as a coffee machine that makes a single cup with a single pod at a time.
(Apparently you can tweak a Keurig to approach an espresso, but it's not the default).
Espresso is so named because you "express" the brew from the beans and you do that with water because water is pretty neutral in flavour, is not poisonous and has quite a few other properties that we have evolved to exploit or live with.
Milk is a weird liquid associated with mammals nursing infants. We humans have evolved to be somewhat lactose tolerant post infancy which is rare in animalia (1)
Given that we are using the Italian word - espresso - then let's use their definition. If you add milk then you have a latte or a cappuccino or an americano con latte, a flat white or whatever.
Real weirdos try to milk oats. I've tried but I can't find their teats.
And lactose free milk is a thing, for those of us who can't have lactose anymore.
as in, a glass of just milk?
FWIW, once you poor milk into your espresso, it stops being an espresso, and depending on the coffee/milk ratio (and milk preparation), you know have a Cortado, Cappuccino, or a number of other versions.
So ultimately: no, no one drinks espresso with milk, it's not a thing, they start with an espresso, then turn it into a different espresso-based coffee drink :)