Yes, you should have a label on your luggage when traveling in the TGV. It says so on the electronic/self print tickets as well as on the ticket sleeve when you buy your ticket in the station.
If your luggage is next to you, you can get away with it not being labeled. But if it is at the end of the carriage, or you walk away from your luggage just at the time they come through the train to check, the staff will ask who's luggage it is. If no answer is given, (or if they are strict on the rules) the luggage can be taken away and destroyed.
I bet they are very careful after a few terrorist tries to attack trains.
If you have a few minutes in the station before getting on the train, you can get a label at the ticket window, likely you can just walk past the queue and get one.
In case of need, just a bit of paper with your name and your (mobile) phone number will do.
(You can even slip it in a pocket on the case or in under a strap around the case, as long as a bit sticks out and your write enough on it that they can recognize it as a label.)
Many years ago I was in a TGV and the staff was asking about luggage, in French, very fast spoken and not over loud. The owner of the case had not understood what was going on and was given quite a bit of talk. (I do not remember whether he understood French or not, but even not speaking French I understood the guards were not happy.)
That was in a time when there were bombs going off in cities, not in trains, but even then labels were needed.
I have just checked the TGV provided label on my case and the only information you are asked to fill out is 'name' 'first name' and 'telephone number' in French. Showing they are not interested in knowing where you live and where you go, just how to contact you when they find your case without you near enough to answer their questions.
I guess that if you do not answer soon enough, your phone number will be called, or you will be addressed through the train speaker system to return to your luggage.
And if you do not respond to that, they can/will take further action.