The US dollar is a medium of exchange. It certainly is currency. But there are a few reasons why it does not satisfy the definition of money.
The US dollar suffices as a medium of exchange, but as Mike Maloney explains, the US dollar is not a store of value - it loses purchasing power over time. Therefore it is not money.
The US dollar is not lawful money. It may be legal tender, but it is not lawful money. The US dollar was backed by gold until August 15, 1971. It may seem hard to believe but you used to be able to walk into a bank and exchange a dollar for a gram of gold.