This is just a data-point from my own background. I double-majored in math and physics as an undergrad and got my PhD in physics. I did a postdoc and then became a community college teacher for 25 years. I taught 98% physics and 2% math.
I'm pretty sure we never saw LU decomposition at all in my sophomore course at Berkeley. I think I learned it for the first time in an upper-division course. (I think we may have seen it both in upper-division linear algebra and in numerical analysis.) If you asked me today, I would not remember what it was for or how to do it, just that it's the name of some way of factoring a matrix. The fact that it can even be defined for a non-square matrix is a complete surprise to me.
Speaking purely as a physicist, we basically never use non-square matrices. The most frequent application is to operators in quantum mechanics, which are always maps from a given space back to the same space, so in the finite-dimensional case they're always square matrices. In fact, we usually only care about operators that are either unitary (for time-evolution) or Hermitian (for observables).
I don't know, maybe an economist, for example, would have a different point of view, but my impression is that doing LU factorization at all is kind of esoteric for a sophomore class, and the fact that it can be done for non-square matrices is super esoteric.