No, the VIN is usually only put on permanent parts of a vehicle. Catalytic converters usually last a very long time but can eventually wear out.

What's is considered a permanent part of a vehicle? It seems like there are some people etching the VIN on the heat shield but I don't see any evidence of that deterring thefts.

I mean “permanent” in a general sense. It varies by manufacturer. For most common cars, manufacturers put the VIN on a few places on the frame of the vehicle.

I don't see how this could deter the theft, as they are cut open and the shell is discarded anyway

Yeah - the shell / heat shield would be discarded but I wonder if having the VIN on the actual converter somewhere would deter theft.