Why do we draw the $xyz$ coordinate system like this?

I just played around with rotation angles.



Answering the question "Why?", I tend to agree that it's easier, since it's easier to measure distances at least for $y$ and $z$ due to the alignment on grid notebook lines. If it's like in the OPs post, then it's harder, but not too hard though.

The question of how to draw "projections" of 3D objects comes up frequently in engineering drawing and drafting. If you took a drafting class (which no-one ever does, these days) then you'd learn all about this. There is a pretty good explanation on this wikipedia page.

The type of picture you showed looks like a "cabinet" or "cavalier" projection. Both of these are oblique projections (as opposed to orthographic ones), and are therefore unrealistic. People draw these because they are easier, and because they make it easy to measure distances (in one plane, anyway), because there is no foreshortening.

Nowadays, 2D projections are usually produced by software, so they are typically more realistic. Actually, even when drawing by hand, it's not all that difficult to make things more realistic. I'd say your teacher should practice a bit -- anyone who is teaching 3D geometric concepts needs to have some drawing skills.

In 19th century mathematical writing, we more commonly see the axes pictured differently...


We are outside the first octant, looking in through the XZ plane.

Even today, you will see this in some engineering or physics texts.