IMO there's no correct answer. It depends on the purpose.

Intuitively XY forms the plane where things are and Z is used for the "additional dimension". Since first we think of 2D, and then we move to 3D, so it's the natural way we think.

So if you make a cities skylines type 2D game, the XY plane will form the ground. Which means that when you want to make the game 3D, Z means up.

But if you're making a platformer game (like old marios), you have XY being the left-up plane, like when drawing a graph in a paper. Then if you want make your game 3D (as in, having 3D models for assets, the movement would still be the same 2D). Then the Z will be forwards.

Sometimes, even Y being down makes sense. Like with UI programming. This is specially important for window resizing. When you resize the window, you want everything to look roughly the same, but with more space. That is easier if you use (0,0) in a point that doesn't move when resizing, like the top-left corner. Since the bottom-left corner moves if you make the window taller.

In Physics we mostly used right-hand, but X-right, Y-up, and Z pointing towards the viewer.

But that's details. The only important choice is between left- and right-handed, as that affects the signs in the cross product (and some other formulas - generally everything that cares about which rotation is considered positive).

Let's call the axes g o and d.

tbh at this point most people taking a linalg class or getting into game programming are probably very familiar with the minecraft coordinate system

similarly to how minecraft has taught an entire generation basic english

There were discussions recently about the feasability of changing Blender's coordinate systems. The answer was that it was highly unlikely to happen because there was no single place where it was defined -instead, it was assumed throughout the code that Z was up, in probably tens of thousands of lines... and the work to regularize that would be gargantuan with no clear benefit -since exporters and importers are supposed to handle these transformations anyway.

In case of Blender, I'd say, it could be user choice.
But I like Z up right handed because it matches the convention used in high school physics.

Rather, why don't Y-up software change to Z-up, instead?
Were they taught to use Y-up in Kinematics-3D? I doubt that.

Z is depth. Why should it ever be up? Honest question.

I don't think that's correct. Here's a drawing I did when trying to get my head around this.

drawing

I find that trying to make sense of terms like "to the left" tricky when we can rotate the directional cube any way we want. For example, in my drawing for "Y-up, left handed", the red X axis is pointed leftwards. However, we could rotate the unit vector cube so that the X axis is pointed right, and the Y axis is pointing up (i.e. the orientation we're most familiar with for 2D graphs). The Z axis would then be pointing away from us, into the plane of the paper/screen.

In contrast, if we oriented the Y-up right-handed cube in the same way, then the Z axis would be oriented as if to come out of the plane of the screen/page, towards us.

These distinctions only matter when we add a third dimension, so the left or right handedness is basically a question of "when we add the third axis to a 2D square made by the other two axes, does the third axis come towards us or away from us? I apologise if this hasn't made things any clearer — I am able to make things make sense by imagining the rotations in my head, but not everyone is able to visualise them like that.

Okay, am I understanding this correctly? In Y-up right-hand, positive X is “to the left”?

No, the x-axis is usually the only consistent one (it increases from left to right). With right handed coordinate systems you can have Z going into the screen (Y-down) or Z coming out of the screen (Y-up)

From the chart I would guess that in Y-up right-hand positive X is to the right. You look at the palm of your hand.

That way when you develop a 2d platformer you would use a standard XY coordinate system. Switching to 3d would logically add the z axis as depth and not height. A movie is usually shot that way as well.

Of course that analogy breaks down as soon as your base-game is top down. Like a city planer or so.

Anyways, with standards it's often best to just go with what most others do. So kudos to Unreal for not being stubborn.

Haven't really done much in game design, but if I'm understanding right.

In game design, lets go with 2d first. You mark position with numbers. In 2d you have X and Y. So you say to move to 5,5. then from the starting point that will be 5 right, and 5 up.

3D you have the same thing, but a bunch of different groups were working on it in different ways without working together. So you wound up multiple systems that had X, Y and Z meaning different things. In the majority of 3d systems Y represents up/down. (the other 2 directions are still kind of half and half between systems)