There are 3 main ways of reliably creating the dark pupil.
1. a piece of black horn is inlaid into the front of the eye and then polished down…..most common. 90%+
2. a piece of dark horn is inlaid into the back of the eye….rarely done as the effect is not as crisp
3. a hollow is created on the back of the eye and filled with black lacquer or similar…quicker thus cheaper
A even cheaper method is simply to stain or lacquer the pupil onto the front of the eye but this obviously wears off very quickly when handled.
They eye in question is either 2 or 3 above imo
I think we can discount any possibility of injections!
As for how they are securely fitted…..most are simply glued in with a variety of adhesives. Some have a deep tapered reverse to give greater surface area adhesion.
Raising up a metal lip after insertion would likely damage the horn so that’s unlikely and I have never seen it done on organic material, only on other metal inlays
Just for info…..the horn used can be of varying transparency depending on the desired effect….from virtually clear to dark amber to black.
In addition to horn a variety of shell (mother of pearl) can be used for the eyes ……golden lip pearl, black lip pearl, bronze lip pearl etc …..all showing a range of stunning iridescent colours when finely polished. Often mistakenly described as “gem stones”.