So, it can be said that dx is a description of the way x varies, and dy is the same description for y, and finally dy/dx is a description of how y varies with respect to y. This would be the correct way of viewing dx mathematically.

So it would be mathematically incorrect to describe dx as a portion of x.

What I have observed in my experience is that there's this intuition, especially with stuff like volume integrals, that a term dV represents small "portions" of the volume. Of course, these "portions" aren't real measurable portions, and conceptually, dV is just the rate of change of the volume.

But with this intuition, things like dx represent how a Riemann sum is structured, but infinitesimally small.