Aight folks, let's break this down.
As someone already left a chef jean pierre onyo image saying they are always number first, let's talk about why.
It comes down to taste, mainly. Texture is second.
Onions contain a lot of great sugars that need time to show up, and some more pungent flavors that need time to break down.
They also start out crisp, which is not usually desirable in the kind of applications this is about. If you're sauteing onions and mushrooms and wonder about the order, you aren't making a salad, right? It's a sauce, part of a soup or stew, something like that.
You want the onions to soften up, and go past the point where they can be a little tough. A tangent on that: when you're prepping onion and remove the skin, also remove the first layer. It is more fibrous, and stays tough-ish even when heavily caramelized.
There are cases where onion can actually go in at the same time as other ingredients, like making stock, or even when being used as part of one of the culinary trinities. But, they will still have a sweeter, more complex flavor if you give them a little extra time on their own first.
Mushrooms, the other part of this meme also benefit from being an early ingredient. As you cook them, they give up their water. This intensifies their flavor, obviously, and it makes the end result more controlled. But, past maybe a few minutes, they don't need more cooking. You can cook them for a long periods as long as you control your heat source and don't burn them, but it isn't going to bring much to the table once you've got a little extra brown on them.
So making them number the first does nothing beneficial in almost any case, but does make it harder to monitor the color and progression of the onions.
Again, there are exceptions. And, if you prefer your onions more pungent rather than as a layer of flavor that supports the rest of the dish, you aren't obligated to follow the principle. I tend to prefer just doing a second batch to put in towards the end, if that's my goal though.
As a rule of thumb, you start with fats. If you're rendering something, like bacon, that's first. But be it oil, butter, or something like lard, that fat isn't just for flavor, it helps the cooking process as well as being a lubricant.
Onion will come second. There's no real use case for onions being in a pan with heat without a fat.
If mushrooms are being used, they'll be third because they need time to release water and develop flavor. Add them in when the onion is just going translucent. They also absorb some of what's in the pan before them, so you'll get all that onion sweetness and depth into them, amplifying their earthiness.
From there, it tends to be garlic, if garlic is used (and if you aren't a vampire, you should be using it), but it can be whatever it is the recipe calls for.
With garlic though, you really want it to be the thing you add right before any stock, water, cream, or other liquids. This is for stovetop applications mind you, and it assumes the garlic isn't whole as a flavoring agent that can be removed. You put the garlic in, and move it around, until you just start to smell it in the air. That's what the whole "until fragrant" means.
As soon as you smell the garlic coming off the pot or pan, you add your liquid, and all those volatile compounds stay in your food insist of burning off.
If you like less present garlic, you can go past that point as long as you keep the heat low, because you don't want the garlic to brown much at all, barely golden. It's hard to visually track garlic in a pan full of onions and mushrooms, unless the heat is low enough to catch it. You start getting brown garlic, it turns both bitter and unpleasantly firm. But, I gotta be real, if you're not wanting that garlic pop, why add it at all? You don't have to. Almost all ingredients can be optional unless you're baking (and even there, there's wiggle room). Garlic cooked past fragrant in a stovetop application doesn't bring anything that the onion doesn't.
Now, if you roast the garlic, that's a whole nuther kettle of fish. It's also amazing. It just needs to be added towards the end of whatever you're cooking, so that the flavors roasting it brings don't get diluted.
Again, there are exceptions to almost any food rules. We're talking general purpose stuff here.
