Such an ID app would have a major advantage over physical wallets: If your smartphone is lost or stolen, you can log in to your account and invalidate those accounts on that device, then transfer them to a new one. That capability already exists today in Apple's Find My iPhone app, which has been copied by third parties for use on other devices. Find My iPhone can locate and then lock or wipe a missing device, but it's not a stretch to see that it could also deauthorize that device for credentials stored in the ID app.
Credit and debit cards are really no different than any other information card, at least in the United States. The PIN is not stored on the card, and the card doesn't do anything active to prove its identity; there are no smarts on the card. But there could be on a smartphone version, such as alerts when you've reached specified balance thresholds.
Transit passes and the like are stored-value cards, where the card's data is updated each time you make a transaction. The reader not only reads your current balance (and account information, so it can log the transaction for you to see in a printed or online statement) but updates that balance by changing the information on it. That's a bit trickier than information cards, but not much.
NFC would allow the smartphone to receive new data, as would some forms of RFID chips. A scan-oriented reader, such as those that read QR codes, could display a QR code after the transaction is complete, and the smartphone app presenting its QR code could read the reader's QR code and update the smartphone's stored value for that account. But that's clunky and requires pricier readers.
A simpler option is to transmit the new balance to the smartphone over 3G or Wi-Fi to update it when the smartphone has Internet access. That can be less than real time in some cases, but it works well enough in many transit systems that had to account for times when the train or bus reader was not connected wirelessly and later update the electronic transit pass.
Cash could be handled as any other type of stored data, such as a transit pass. The balance data could be stored on the smartphone itself (better for those fearful of having their finances tracked) or transmitted to a server, as is the case with those prepaid debit and credit cards. Like real cash, if you lose your smartphone, you lose your cash.
In this day of e-cash (debit cards), printed cash seems antiquated. I've long thought the government should offer us all cash cards that we can recharge electronically, but that hasn't happened because Congress doesn't want its friends in the banking industry to lose all those processing fees. The barrier here is political, not technical -- and I bet it would be cheaper for both the government and retailers to move to e-cash than to keep handling printed cash.