As far as I know a horse can be registered and not tattooed.
Both mine are unraced and not tatooed. But both are also registered with the JC. They usually tatoo the day of their first race. At least this is what my Dad (a TB breeder) has told me.
Someone please correct me if I am wrong. It could vary from state to state...not sure.
It is totally amazing what one has to do when they pick out names etc. The person has to submit more than one name and it then has to go through an approval process.