Is canned “tomato sauce” already cooked?

Yes, in some sense, the contents of a can of tomato sauce are already cooked. But "cooked" is a pretty vague term, and we know that there are different degrees of it. Did you cook your onion on a low heat until it was soft, or until golden, or until caramelised? Did you cook your steak until it was rare, or medium, or well done?

Likewise with tomatoes, if you blend them up and then cook them for ten minutes they're not going to taste the same as when you've cooked them for three hours in a low oven with the lid off. Lots of chemical reactions are going on inside the food as it's cooking, and some of them only happen over rather long time periods.

So what you're doing when you're putting the "cooked" canned tomato sauce in with your "cooked" meat and leaving it for a while is... cooking. The combination of ingredients plus more heat and more time will cause further changes and different flavours to emerge.

Also, canned tomato sauce will only be cooked enough to preserve it in the can - it's not been cooked to the point where it's going to be at its best for eating. That part's up to you.

The term 'cooked' is being used subjectively here. Canned tomatoes are 'pasteurized' which is a process similar to cooking. The contents are held at a slightly elevated temperature for a short period to kill the germs. The temperature and time used are not enough to significantly change the structure of the contents, however it may slightly change the flavor. This step of the canning process is necessary to avoid botulism and other bacterial growth while the canned goods are waiting to be used. You can use the tomatoes straight out of the can if you wish, provided that the canning process was performed correctly and the container is not breached(dented, rusted through, punctured, etc). However if you choose to cook them further you will most likely be using temperatures much higher than the temperature used to 'pasteurize' them and the structure and flavor will change significantly. Tomatoes are rich in nutrients. Cooking the tomatoes further changes the nutrient content and effectiveness. Lycopene for instance is activated by cooking the tomatoes. Vitamin C however is destroyed by the cooking process and possibly by the canning process. i.e. Fruit juice that is pasteurized has to have Vitamin C reintroduced. (sorry for the sideline but I feel it helps to understand the difference between cooking and pasteurization).

Did you first brown your meat before adding tomato sauce? If you did, there's a chemical reaction called the Maillard reaction between the sugars and protein in the meat. The byproducts would mix with the tomato sauce altering the flavour. If you used onions and browned them, they too would change the flavour. Even spices will mingle to make a new combined taste. Don't forget that tomato sauce is acidic and acids de-nature protein (break them down) in meat, some which dissolve into the sauce enriching its flavour.

If you were to pre-cook sauce and meat separately, then mix them (as Megha mentioned), it wouldn't taste the same as if cooked together. Heat speeds chemical reactions up and cooking is a chemical reaction between different foodstuffs. If you were to mix them pre-cooked, it's conceivable that it may taste similar to cooked together if you allowed enough time for the reaction to occur. But it would proceed so slowly at refrigerator temperature (or even room temperature) that the food would spoil first.