If an unlicensed operator is operating phone on an amateur frequency under the direct supervision of a licensed operator for, say, longer than ten minutes, should the unlicensed operator say the call sign of the licensed operator, or should the licensed operator take the mic and say it? This is not like in Field Day or a special event station, which are covered under a different rule [97.119(d)].
Yes - they can ID your station.
Part 97 does not specify who must identify your station, rather, that it must be done in 10 minute intervals.
As a licensee, you are ultimately responsible for all transmissions that come from your station. If there is third party traffic (your friend talking), this means you must be present at the control point to disable the transmitter should any violation of part 97 occur.
You must be in control of the equipment at all times, and are still responsible to make sure that your station identification is given per part 97.
Short answer -- anyone can give your call sign, as long as it is done, and a licensed ham is the control operator and you authorize the use of your call sign.