The TGV has mandatory seat reservations (unlike, for example, the ICE). There is no equivalent of German Flexpreis where I can take any train, and if I miss my connection I need to get a new ticket issued. Yet on the TGV, there are often small foldable corridor seats, which do not have seat numbers assigned. When travelling on TGV, I've often seen those in use. What is the purpose of those seats? My impression is that in case of disturbances, the mandatory seat reservation is not well enforced, but I don't know if that is at the discretion of the train manager or if there is an established policy.
The TGV has mandatory seat reservations
That is incorrect, e.g. see https://en.oui.sncf/en/france-europe/on-board-seating-positions (mirror):
When all seats are booked, you will be offered an "no fixed seat booked" seat. In this case, your ticket will have a coach number and be marked "sans place attribuée" (no fixed seat booked).
Rest assured, to ensure we continue to deliver a quality service, the number of "no fixed seat booked" seats is limited. When you board the train, locate the ticket inspector who will help you find a place.
In practice, it might not be practical to locate the ticket inspector. If you have a "no fixed seat booked" seat, you simply enter the train and sit wherever you want as long as 1) nobody has booked your seat and wants to sit there 2) the ticket inspector doesn't ask you to move somewhere else. As a result, you may end up on the corridor seats, and sometime on the floor. But not on the roof.
When I missed my connection I went to the ticket window, I was accepted on the next train even though, in theory, there were no seats available. I had to report to the train manager. He asked me to sit in one of those extra seats till the train was well underway and then move to any of the empty seats, as they expected other people to miss that train.
I have heard alike stories of other people, some of whom could not find empty seats and stayed in the corridor seats.
Note that the rules have recently changed, are still in flux, and of course things may be very different in case of disruption.
- On TGV seat reservation were mandatory
- There would be a small number of reservations made without an assigned seat, and no guaranteed seat. This is just plain and simple overbooking. In many cases there will be no-shows and you will find a seat anyway, in others you'll just have to sit on the "strapontins" (foldable seats on the door platforms in non-Duplex TGVs) or "banquettes" (side seats on the top floor of Duplex TGVs). Or standing at the bar.
- But there was quite a bit of flexibility for some types of tickets and frequent travellers, which meant they could just hop on the train before or after the one they had booked. The rules were complex (depending on your category, it could be only trains before, or only after, it could be any train before or just the one before....). It at least some cases, you were supposed to see the train manager when boarding, and they could accept/refuse you on board.
- The ones "abusing" this the most were the holders of a "Forfait" (pass) that allows them to travel as much as they want while paying only a nominal amount per trip (I believe it's 1.50 euros or something similar), as long as they reserve, but they could change to earlier/later trains. This meant that even if the train they wanted was full, they booked the train before, and then got on the train they wanted.
- Some categories of travellers had the "reservation train complet" privilege which allows them to change their reservation onto another train the same day even if full.
So, as you see, there are lots of possible reasons there could be people standing or on the "extra seats" in a TGV. Even with all regular seats occupied, you'll find the bar coach quite full on some trains (even if the bar itself is actually closed for service), especially the "commuter" TGVs (Lille-Paris, Reims-Paris, etc.).
A few weeks ago they changed the rules a bit, and now you supposedly can't just hop onto another train. You have to change your ticket before boarding. However, there are still plenty of people who still have all the same privileges as before, so it probably won't change things much (maybe they'll put a cap on the total of "extra" passengers? I'm not sure what the goal of the change really is).
I haven't had the "opportunity" to exercise any of those privileges since the changes, so I can't really say how in practice this affects things.
Note that beyond a certain number of passengers (but this only happens in cases of disruption), the TGV is not allowed to travel at full speed, so train managers try to avoid getting to that point.