I saw a lot of recipes asking for cooking meat with other ingredients such as onions. I thought the reason behind it was that the amino acid in the meat can work with sugar in other ingredients to facilitate Maillard reaction.
Is that correct? or that is not the reason?
Is Maillard reaction only related to the amino acids and sugar in the ingredient itself and not related to other ingredients?
I wouldn't say that it is "because of the Maillard reaction", or at least not in the strict either-or sense you seem to assume in the question.
There are thousands of reactions which happen while cooking, and only some dozens of them belong to the Maillard family. And certainly a larger variety of reactions happens when you have more ingredients in the pot - simply because you have a larger variety of inputs to react with each other.
Thousands of years of observation have found out that this results in tastier meals with less effort. First, it increases the complexity of the taste you create. Second, you want to brown the other ingredients too, not just the meat. Given the choices of 1) brown the meat and not the others, 2) brown each ingredient separately, and 3) brown all ingredients in the same pan at the same time, you will find out that the third option is the easiest one logistically, and gives you a somewhat better taste than the second, and a hugely better one than the first. This is why cooks do it that way.
The kernel of truth in your assumption is that indeed, Maillard reactions happen there. They happen both when you cook the meat alone, and when you cook it with other ingredients. It is even plausible that different reactions happen when you cook it alone than with other stuff (e.g. meat protein reacts with meat sugars, vs. meat protein reacts with onion sugars). But that's only a tiny fraction of what happens when you cook the food. So I would not declare it "the reason". The reason is the total combination of all these not-yet-researched reactions that happen when you brown meat and vegetables and seasonings together, which chefs like to think of as the art and magic of cooking.
That's correct. Maillard reactions occur between reducing sugars and amino acid-containing compounds (index of multiple articles). The 'sugars' naturally present in meat are stored as the polysaccharide glycogen, analogous to starches in plants, and are not readily available for use as sugars.
Onions also contribute sulphur compounds that take part in reactions (The Chemistry of Meat Flavor - beef industry sponsored). Some more info here: What happens to the flavour of meat when you cook it Star Anise with Onions?.
(There may be a language-related point here.) Maillard reactions typically take place at 140 °C or hotter. That's frying or roasting (or baking), but not cooking as in hot water or steam at ambient pressure.
Nevertheless, cooking of meat with onions (or other veggies) is a good idea e.g. in a stew since the combination of (cooked) onion flavor and meat flavor is pleasurable to many people.
(In particular I guess many people would say that the flavor of some veggies improves a lot by having meat flavor in ;-) )
Whether there are some additional flavor-important chemical reactions that occur only if meat and onions are present in the same broth during cooking I cannot say.
The control experiment would be to cook them separately, let cool, combine and let sit long enough for the dissolved flavors to diffuse into the other pieces.
There's also the indication that many people consider it sufficient to use broth powder or canned (i.e. already cooked) meat for every-day meals and prepare stew from scratch only for special occasions.
