Obviously Rand is referring to Bitcoin the technology generically. Based upon his usage of the terms we can call our 'shares' bitcoins just like we call all tissues Kleenex or copiers Xerox machines.
Rand Paul's reasoning is suspect here. If Bitcoins are superior to traditional stocks then who does it benefit to back something superior by something inferior? It's like saying e-zines should be backed by print magazines or email should be backed up by print copies.
He wants Bitcoins to be backed by stocks then shouldn't he be talking to his friends to get the SEC to allow equity crowd funding?
The law is already passed but the SEC is delaying putting it into effect. I don't think Bitcoin itself could be backed by stocks and I don't think people would want that.
In some ways Bitcoins are a stock but designed to be independent of governments. A Bitcoin using Colored Coin could in theory be redeemed for a stock, or using Mastercoin, but by backing it by stock it puts the entire Bitcoin ecosystem under the control of politicians rather than programmers.
If we just have Bitstocks or Bitshares which are backed by smart contract then in theory you can have a Bitshare asset backed by equity in the real world, but why call say it's backed by a stock? If you can do the whole thing digitally why would you need it on legal paper at all at that point?
I see the whole "backed by stock" thing as a trojan horse to get Bitcoin, Bitshares and everything like it to be backed by the law, and if it's backed by the law then it's backed by the signatures of politicians who write the laws, as well as the quantitative easing.
Maybe this is a bit of a stretch, and may be seen as a conspiracy theory, but it's also possible that Rand Paul had good intentions and just didn't understand how the technology works.