Yes, his hole is indeed a singularity. The hole in his face isn’t a black hole, that’s the hole he uses to blast energy. The black hole is at the center of his chest.

Because Copoctatius is an enormous black hole, he cannot spaghettify objects since the tidal force is the same on the feet and the head.

His skin is made of an extremely rare material, even objects such as black holes cannot destroy it. So, although while his body wouldn’t be destroyed by the black hole, he would suck some of the bright energy on his hand off. But the source is practically endless and is extremely slowly diminishing so it would easily regenerate.