I broke a piece off of a cultured marble vanity when removing a
faucet. Can this be repaired?

This material is called "cultured" to make it sound real high toned, like
cultured pearls which are farm gown real pearls. The so called cultured marble
is just a resin and marble mixture cured in a mold. The resins are often
polyesters, so any material good for repairing a boat would work just fine.
Most serious repair people would use a two part epoxy for this job. HTH

Joe