When I play a game that is not a teaching game I or my opponent might say "Shall we go over the game? We then replay the game and discuss situations which we feel are interesting, e.g. investigating possible move improvements or variations. We go on with this as far as we want, often reaching well into the endgame. Sometimes the review takes longer than the game, even when the game was a slow one
When I am the receiver of a teaching game it is usually with a pro and what happens depends on the time available and whether I am paying or not. If I'm not paying, e.g. a 3-on-1 at a congress, I would expect the pro to point out my five or so worst mistakes and a better way to play in each case. If I'm paying, it depends on how much time is available. Usually the pro decides what to tell me or work on with the time available. If it is a regular lesson situation then there may be suggestions for what to work on until the next lesson. If I'm giving a teaching game, I usually pick out several serious mistakes and discuss them in some detail, such as topics like keeping your stones connected, keeping away from strong groups, not using a weak group to attack, making your stones work together, making your move work efficiently, tenuki, balance, etc.