In Ohio, as long as he sticks to the basics, like his website implies, he doesn't need any sort of license. As far as plumbing, he could repair a leaky faucet but isn't allowed to install new plumbing in a bath or kitchen or tie stuff into the sewer system. If he does electrical, he could replace an existing light fixture, he cannot re-wire a house. I think, in general, a handyman can do stuff that doesn't require a permit.
It looks like a general handyman can get liability insurance for about 30 bucks a month although I don't know that the state requires it if he's not licensed. If he does mess something up and a customer sues, he has no assets and he's a legal adult. There's nothing for a plaintiff to recover. No lawyer is going to take the case anyway.
A handyman who does more involved jobs would need a general contractor's license which would require at least 3 years' experience working with a general contractor. I could see David and Uncle Kev exaggerating Timbits' credentials to get him a general contractor's license.
Timbits might've made sure his religious beliefs were front and center if only because these fundies tend to be a tight knit bunch and someone with likeminded beliefs is going to be less likely to sue for defective work and also more likely to allow him to do work outside his wheelhouse without a permit and pay him more than he usually makes while not as much as if they'd hired a real pro.