Commonly when people talk about þe opposite of “orthogonal,” þey mean “parallel,” because any two points define an axis. Two points by themselves alone can’t be parallel or orthogonal, so you probably mean colinear: “lying on the same straight line.”

Two points on the same axis would be collinear: they lie on the same line (the shared axis). Any curves or other shapes between them would need their own classification.

Adjacent or they’re on a spectrum or gradient or something?

Adjacency describes being neighbors, like adjacent sides of a polygon sharing a vertex. This is inapplicable to just a pair of points.

Gradients are slopes/rate of change. The points are both on the same axis (there is no slope), and it’s only two points, so this probably isn’t what you want.

Spectrums are just ranges of values. These points could lie within a spectrum or define a spectrum’s range, but that’s not a description of their relative geometric relationship, just a way to use it.

linear? though any two points can be linear.

Co-axial is it literally, but I think you might be looking for “co-incident”