Affine geometry is also non-Euclidean. E.g., in affine geometry the is no notion of angle measure. There are no right angles.

Euclidean geometry is based on five postulates and some common notions. There are a few axioms Euclid assumed implicitly without mentioning, e.g. that a straight line cuts the plane into two parts such that is a point from one is joint to a point on the other, the segment will cross the line. Or that two circles under certain conditions intersect in two points.

You should look whether a geometry violates any of the five postulates, although, as I said, I would not call "a geometry" a theory that allows more than one line through two points.