Does shearing a sphere generate an ellipsoid? In my preferred 3D modeling software I see that however I shear a sphere, I seem to be able to make a nearly identical shape using some combination of non-uniform scaling followed by rotation. Are these indeed the same shape, and if so what is the mathematical relationship describing how much scale and rotation will match a given amount of shearing?

Yes.

Your sphere is of the form $x^2+y^2+z^2=r$, now after shearing the quadratic form $x^2+y^2+z^2$ you will still have a quadratic form and it will still be positively definite (obviously).

Now for any positive definite form you can identify eigenvectors (which will become axes of the ellipse). Changing to the basis of eigenvectors the form will take the form $\alpha u^2+\beta v^2+\gamma w^2$ and the equation of the sheared sphere will take the form of an ellipsoid $\alpha u^2+\beta v^2+\gamma w^2=r$.