What will happen if you cook in a container that has traces from another day of cooking?

There is no food safety issue involved, but in most cases, there will be a problem with taste. I think that the most important factor is whether the recipe involves heating (in the container in question) or not.

If you must heat something in the container, don't reuse. Different parts of a heated recipe are added at different times (e.g. for a stew onions and tubers first, then meat, then fluid, then the delicate veggies, and herbs at the end). Having some pieces of the old vegetables at the bottom will interfere with that. Even if you don't have such a dish and are heating everything at once, you will cook the old food for the second time, overcooking it. It will taste bad, or even burnt - not only the old pieces, they will give their bad taste to the rest of the dish.

If you don't heat in the container, but only do some physical changes (mixing, whipping, blending), it will probably be mostly OK. Take care with dishes where the mixing order is important - if you made a soufflé mixture in a bowl, you cannot start another one in the unwashed bowl, because you have to beat the egg whites in a clean bowl. If you get impurities into the eggwhites before they are beaten, they won't aerate properly, even if mixing in the same stuff after they are stiff presents no problem. For a chemically leavened (baking powder or baking soda) batter, you cannot start the dry part in the bowl which contains moist batter traces. And if you made anything which ferments, you will have some overfermented fragments. I wouldn't do it with yeast containing recipes, but if you are making yoghurt, you can just see it as a part of your starter culture.

If your food doesn't belong to these exception categories, you are probably OK. Still, your food will get some off tastes. Even if you remove the last of the old food immediately before the new cooking process (and do you really start cooking just after you've eaten?), the old food will have left dried traces on the walls of the container. These will mix or dissolve into the new food. The drying alone will have given them an unpleasant texture. If we are talking pieces, or even worse, purées of fruits or vegetables, they will likely have oxidized as well. Anything which changes its taste with age (like fish) is better washed away.

So much for the downsides: not game-breaking, but they exist. On the other hand, I cannot see any big gains. Cooking gets much more utensils dirty than the container in which the food is stored/prepared. If you are already doing them in the sink, elbow-deep in soapy water, the marginal effort for rinsing an additional item is close to nothing. And if you have a dishwasher, I don't see any reason to not use it. Conclusion: there is no reason to do it, even though it won't poison you if you do.

Nothing, except maybe a trace of taste if it's something strong.

You probably shouldn't do this with frying pans, though; I think if there's anything stuck to the bottom it will make your oil taste burnt, especially things like garlic and onions.

Besides mixing flavors there isn't a problem with reusing a pan for a different dish right away. My cast iron pan is often used this way.

As for storage in the fridge; sure you could probably reuse the container after one day with little risk. Two days? Maybe. A week?

The point is that it is a bad precedent that will increase your risk of food poisoning. Are you going to label the container so you don't forget and reuse one that was already reused? Remember that bacteria is growing all the time even in the fridge. Reusing a dish that already had a chance to grow some colonies is going to give the next dish a head start on spoiling.