"Take any open seat": get a boarding pass!
I got on an earlier flight to ORD last week after seats opened up at the last minute, and the GA told me to take any open seat. I found out later he didn't check me into that flight. As a result, when I tried to check-in online for the return trip, my reservation had been cancelled. I called reservations who promptly re-activated it. However, I had to write to Mileage Plus to request credit for taking the standby flight.
This is the second time I've experienced this. Not only is it inconvenient, I see this as a safety issue. You're on the flight but not on the manifest. Heaven forbid, in an emergency the airline would have no record of you on board.
Moral of they story: if you get on standby, make sure the GA gives you a boarding pass or some documentation that you're on that flight.