Can The Government Identify What You Bought With Your Credit/Debit Card?

The government cannot tell from your credit/debit card transactions alone what you bought. They can see:

• When you bought something
• How much it caught
• Where you bought it from

Essentially this is what you see on your credit card statement. This is often done through a warrantless "hotwatch" program that can be done in real time without your permission or knowledge. It is often done to track criminal or terrorist suspects. A court order (subpoena) is required and it is not just an automatic thing like running a license plate.

Merchant Category Codes (MCCs) do identify what kind of business you've patronized. For instance, if you bought a gun the credit card company will see MCC 5941 Sporting Goods Stores. From here they can draw few inferences that you may have bought guns or ammo. 

Knowing where you shopped means that investigators can obtain any internal records from the business, such as your purchase history. This could be tracked by your Amazon order history, a shopper's club card (they use your purchase data in exchange for 'discounts'), or a membership card purchase history. In this case, they don't even need a credit card as long as they know where you shopped.
​
There is no way that I'm aware of that purchases can be tracked and identified in real time. A lot of legwork is required and it's impossible to do for everybody. Although technology is changing rapidly and so will tracking in the future. 

I stopped using my club card, phone number, or email at sporting goods stores. ​So if you want to truly remain unknown, pay cash, use no phone number or card, and buy at a store far away from home where no one knows you. 

Update: The MCCs are only part of the story. "Level 3 line item data" may be be voluntarily sent by retailers in Level 3 data because it lowers their processing rates. IE if we know what your customers buy, you get a discount. Probably many retailers have this on by default and don't even know it.

"Level 3 processing and sending line item detail is an advanced method of processing commercial, purchasing, fleet and government credit cards. It involves sending additional fields of data through the processing network. This is up to 12 fields of additional information, much like you would find on an invoice. With the additional data a business can monitor what kind of purchases are made on a company card, where, how much etc. It can also include restrictions, limiting the types of businesses an employee can make purchases. Visa and MasterCard created special rates to support Level 3 data by reducing the transaction cost when level 3 processing data is included with the transaction. This often lowers the transaction cost by up to 40%."