Drilling Through Faux Marble?

In all probability, it is polyester resin. Easy to drill/cut. Just drill a small, shallow starter hole. Or even use a knife to dig out a small & shallow cone.

My guess as well. Treat it as if were wood, basically. Use a sharp center punch or ice pick/awl to make the starter hole and start first hole with small diameter bit

There is enough aggregate in that matrix to dull a regular drill pretty fast but if you are ready to chuck it when you are done, it might last long enough for a couple holes.

...but don't rap the center punch with a hammer, just use it scratch the surface enough to remove any glaze. That will help prevent the drill bit from walking.

I would also consider drilling very slowly and maybe even using a little bit of lubricant. Depends on how hard the material actually is.

Do you know where the studs are?

You won't need a special drill, but you will need a masonry bit. They aren't particularly expensive.

Put some masking tape where you want to drill, don't use excessive force, keep the area wet with a squeeze bottle. There's plenty of Youtibe videos.

Dip the end of a soda straw into epoxy, liberally. Stick that end onto the surface where you'd like the hole. When epoxy has cured, cut straw off. This leaves a ring of epoxy adhering to the surface. Center drill bit in this ring. Proceed slowly.

I haven't seen miles of "faux marble" but none I have ever seen has any aggregate. Color (swirled into the resin), yes; aggregate, no.

I know "Cultured marble" will destroy a regular carbide saw blade. I never drilled it. You may be thinking of something else I guess.

'Faux marble' usually refers to a finish on a wooden surface to resemble marble. The molded man made marble used for bathroom sinks and so forth is essentially fine sand in a polyester matrix.

Per (PeteCresswell):

Dremel tool with the little burr-like bit:

Coincidentally the right hole size too.... Went through the faux whatever-it-was like a hot knife through butter. I stopped when it hit the wood, and then followed up with a regular wood drill.