how strict are seat assignments on pre-booked trains?

You can of course occupy any seat that is not labeled as reserved, no problem.

Jen,
German: Wagen = Coach/carriage/car/vehicle, Platz = seat, as in example here: http://sampaisudah.com/wp-content/uploads/2014/10/DB-Bahn.jpg

It depends on how full the train is. DB (German Railways) have a rule that if a set is still unclaimed 15 minutes after the train has left the station, anybody can use that seat.
German trains have a little indicator above the seat, which says the stations between which it is reserved, for example "Saarbrücken - Paris". Just because a seat is empty when you leave Stuttgart, it may be reserved from (for example) Saarbrücken.

I would say it is less lax than Italy. And French High Speed trains are reservation only, average about 80% full in 2nd class. Keep your fingers crossed.

I would start out sitting in your assigned areas, with one adult in each car, even if that's not how you're named on the tickets. When the conductor checks tickets, you could ask about opportunities to move. Or you may even see a conductor outside the train as you board. You could also ask other passengers about making a specific seat swap to their reserved spots.

Several years ago, I was in a kerfuffle on a French TGV from the Spanish border near San Sebastian, when the conductors made everyone go to their assigned seats, but that was because an unusually high number of passengers got on without the required seat assignments (or perhaps even valid tickets). I don't think that will happen on a TGV departing Stuttgart, where the reservation requirement should be well known, by now.

I'm not sure why your Italian train experience seemed rather lax. Trains there tend to either require seat assignments for everyone or not offer them at all.

My experience is that Germans tend to be somewhat insistent on following rules, so if their tickets say to sit "here" they sit in that spot. Many frequent travelers also have specific seats they prefer (Tables, direction of travel, outlets available, etc.) so they will be especially insistent that you move. Seats will be labeled with the reservation info, so if a seat is not occupied for that leg, you are free to sit, just be prepared to move when the reserved leg comes up.

My experience is that Germans tend to be somewhat insistent on following rules, so if their tickets say to sit "here" they sit in that spot.

I think it's quite normal to take the seat you have reserved and paid for. You are, of course, not obliged to occupy that seat if you like better another free one. And you can take of course any seat that is not marked as occupied. However, if I have booked a special seat (e.g. with a table, since I want to work) and have paid for it I don't want to see it occupied by somebody else at my expense. And if I have taken inadvertently a seat which somebody else has reserved I will move of course myself and not tell him / her to look for a free seat elsewhere.

@Jen: You can change seats assigned by the system at no cost once. You can do that online; details here. In case of problems go the InfoPoint in Munich Hbf.

We reserved specific seats in a train compartment from Frankfurt to Hamburg (6 seats facing 3 to 3), with our names on the digital screen outside compartment. When we got there our compartment was full and the family looked annoyed that we were in their space. I had my printed info from the bahn.de site in hand and told them in German that we had two reserved seats. They left but not without giving us dirty looks.

My experience is that Germans tend to be somewhat insistent on following rules, so if their tickets say to sit "here" they sit in that spot.

I did not mean to imply that there was anything wrong with sitting in your reserved seat, I certainly would...Just that the OP's experience in Italy would not happen on a German train. Head to your reserved area (at least one of them in this case) and if there are free seats, take them, if not, be prepared to sit in assigned seats.

OP, the second leg, the train departing Stuttgart, is a French TGV train and in my experience quite strict about seat assignments because a seat reservation is required with ticket purchase, unlike the German ICE trains. Everyone will have an assigned seat.

Your best bet is to see if some people are willing to switch, with the help of a TGV employee. But unless your children are really small, this isn't a big deal. The trip is 3 hours, 10 minutes, you can take turns walking to the other car, chatting, going to the cafe car, etc.