Webhosts that Do ask for ID should provide proof they are legitimate registered businesses...
Most would be happy to, I'm sure. The problem in this industry are that the majority of folks are not running legitimate registered businesses, but people keep purchasing from them anyway. Perhaps due dilligence in this case would have been to ask Johnny prior to purchasing hosting if he was a legitimate business. Not that legitimate businesses and business owners don't scam people, but the odds are far less likely. Apparently, it's completely okay to deal with an illegal business, and order hosting from them...until things go wrong.
...and should also request the ID during signup, and not after 2 payments had been made.
As several other people have said, there is no reason to request it during signup. It's not until the order gets reviewed and it looks suspicious or fraudulent. The two payment thing is a bit weird. If it looks suspicious, that would happen generally after the first payment, unless of course the 2nd payment is through a different means, or looks suspicious.
I have had several clients over the years, for example, that have paid 1, 2, even 3 or so times with legitimate credit cards or PayPal accounts, so as not to throw off our fraud systems at the begining. After that, they pay with either stolen cards or stolen PayPal accounts, on subsequent invoices. In those cases, sure, we'll absolutely ask for ID or scanned credit cards on the 4th or 5th payment. I can count probably on my two hands the number of times that has happened, but it has happened. Thieves are getting smarter. The first payment and initial sign-up is always going to be the most stringent, when it comes to doing fraud checks, but that doesn't mean you can just become complacent and accept every payment.
At the refusal of providing ID, a refund should be issued immediately.
Agree 100%. They have absolutely no grounds to keep your money, unless it's something you agreed to when signing the contract. I highly doubt that clause would be in there.