It does, at least in some cases. When I book flights directly from an airline, the bank knows the name of the passenger, the date of the flight, the origin and destination airports, as well as the cabin class. For one bank (Chase), the information is printed on the statements; for another they have a web lookup tool.

Flights are one of the very few cases where this happens automatically though the card providers systems. It's called L2 data, and it's sent a little while after the transaction goes through.