Working in DC, I trip over machines that stamp out dog tags on a regular basis, so I have them made with name and phone number. I've got one screwed to the bottom of each of my Predators.
The same thing could be done zip tying one to a regulator yoke or hose and they tend to be comparatively cheap, rust resistant and durable.
I've seen divers bring in regs for service with engraved round plastic tags (plastic name tag material) with their name on them, an thats the same thing, just in a different format.
The same tags could be used on BCs, but my general concern with tags on regs, BCs etc is that they pose a slightly increased entanglement risk. Not a big issue for a recreational diver, but a problem for a technical wreck diver.
In any event, the last phase in reg servicing a reg is filling out the warranty paperwork for parts replacement and all companies ask for the serial number, so the tech will check and record it. In well run shops the serial will already be on the sheet that is filled out when the reg is received and the tech will ensure it matches the serial on the reg when they finish the service work and paperwork. If not, any mis match will most likely show up when someone creates the invoice for the reg in the system as most shops track regs and serial numbers by association with owner accounts in the book keeping system. Which is to say, a lot has to go wrong for the wrong reg to go home with the wrong diver after an annual service, so a reg is not the first item I'd tag.
BCs will get separated from regs, and alternate air sources and BCs will get separated for service, so I take care to label each item so they are properly re-united later. If you get a large bunch of BCs and Air 2s in for service and DO NOT mark them, the resulting confusion will rank right up there with father's day in Ireland.
Those then would be high on my list of items to mark prior to service.
