Government ID by itself does not prove you are enttled to a government rate or a tax exemption. So the hotel it totally wrong. They copied a document that should be worthless for audit purposes. You normally need travel orders, and any kind of ID would prove that you are the person named on the orders.

Generally, to qualify for tax exemption, the payment must come directly from the government to the hotel. A traveler who uses his personal credit card to pay with later reimbursement does not get a tax exemption. You need a government credit card, in which case the card itself is all the proof you need, and possibly forms to be filled out. And note that DC has a law prohibiting photocopies of the credit card for security reasons.