Similar here. I too have used my CC for refills and may have given my name for caller ID purposes, but I don't think it was mandatory nor do they have my SSN or other hard details on my identity.
AFAIK you shouldn't have to get a throwaway phone either. "SIM Card Kit" or "Bring your own phone/device" may be keywords to research -- you should be able to get a SIM that you can put in any GSM phone, and it will come with instructions about a website to visit, or phone number to call, to activate it. If they insist on names etc and it's prepaid, you could give a fake name since it will have no bearing on your ability to use it, although I'd be worried that it might be illegal for some stupid reason.
AFAIK you shouldn't have to get a throwaway phone either. "SIM Card Kit" or "Bring your own phone/device" may be keywords to research -- you should be able to get a SIM that you can put in any GSM phone, and it will come with instructions about a website to visit, or phone number to call, to activate it. If they insist on names etc and it's prepaid, you could give a fake name since it will have no bearing on your ability to use it, although I'd be worried that it might be illegal for some stupid reason.