It’s like getting a tattoo.
Is it? I’m not certain that it is.
In a tattoo, ink gets injected through the dermal layers. With permanent makeup*, it is applied on the outside of the skin. It may not actually qualify as a tattoo for halachic purposes.
In many ways, it may be similar to eating fake crab meat, which (assuming it is otherwise kosher) is certainly permitted.**
The Wolf
* I don’t actually know this to be fact. If I’m wrong, then obviously you should disregard what I said above.
** I’m not addressing any hashkafic concerns (i.e. is it proper to do so even if technically permitted?). Those are separate questions.