While I'm certainly not an expert on the subject, my vertebrate anatomy background makes me feel that he's correct with the fingernail analogy, especially with "fancy" guppies with large non-muscular tail fins. I'm drawing this conclusion from my knowledge of fish evolution and terminology not from any actual experience.
Almost every fish that is alive today belongs in the Actinoptergii Genus?, group? can't remeber the bigger name, as apposed to Condrictyis (cartilage fish, sharks etc.) or Sarcoptergii (lobe-fined fish, like Coelacanth). The term Actinoptergii comes from the protein thats called Actin the gives the fin rays their stability, as opposed to the muscle and bone/cartilage in the other groups. I was under the impression from some of my biology professors that Actin is closely related to Keratin, the protein that makes up hair and fingernails.
To what extent the fins contain blood vessels and nerves that might cause damage and pain, i can't say, but the main component of the fin rays should react very much like fingernails.
I do think there is another matter that we touch on a lot here, at what point does the fishes well being have to become the priority over our enjoyment? Some people would argue that having fish at all is putting them in danger, others argue very extensive lighting and tanks are absolutely required. On the other end you have Dyed and Tattooed fish and the likes of Glofish, generally disliked things, but not totaly unknown in the aquarium world.
As far as my judgement, i'd say that this is not the worst thing that could be done to the fish, i'd generally say to be careful that your not hurting the fish and keep an eye out for infections, but as far as pain and suffering that we know things like tattooing causes, i'd be unlikely to lump this with that and condem it so harshly