I just looked at the Bahn website, and they are offering the same 59€ Savings Fare ticket price for that connection that Captaine offers.
I think that 8 minutes is enough time to make the connection. Stuttgart is a "stub" station, that is, the trains come in to the end of the track with a connecting platform at the end. Before you get into the station, work you way to the front car with your luggage and be ready to get off when the train comes to a stop. It's a short walk to the connecting platform (no stairs) and then only three platforms between the platform with track 7 and the platform with track 16.
If the incoming train is late, a lot of people will be making the change, and they will probably hold the second train until everyone gets on board. Just stay with the herd. Don't dally. And if the train is one minute late and you miss you connection, you should be able to get on the next train to Munich because your missing the connection was a fault of the Bahn. Just go to the ticket office and show them your ticket. If the incoming train was late, it will show on their computer and they will assign you to the next train.
As I tell everyone, a train is not an airplane. You don't have to remain seated until the train has stopped. Get up before and go to the end of the car, ready to get off when the train stops. If you stay in your seat until they open the doors, the flow of people coming into the train will make it difficult to get to the door to get off.
But that shouldn't be a problem here. That TGV terminates in Stuttgart and the next train from that track is an ICE. There is a TGV (probably the same train) leaving Stuttgart an hour and a half later from a different track, so there probably won't be anyone getting on when you get off.
The Mannheim connection puts you on the same ICE you would get on in Stuttgart.
You can also do a lot with the Bahn website by using the "stopover" function.