Euclides was doing Euclidean Geometry !
If similarity geometry means similarity-invariant geometry, what are its objects? Google has a lot to say about similarity spaces, none of it relevant to similarity invariance.
Sticking to finite dimensions, a Euclidean space is standardly defined as an inner product space over the reals. As such it has predicates recognizing those line segments of unit length, and those lines passing through the origin, neither of which I remember from Euclid, who seems rather to be writing about similarity geometry.
Euclid does however talk about right triangles and line segments of equal length (and equal angles but that follows from the other two). So clearly Euclid is doing more than just affine geometry.
If Euclid's plane is not the Euclidean plane, what is it? If Euclid was doing similarity geometry it should be called a similarity space (certainly not a vector space or a Euclidean space or an affine space or a projective space or a metric space or a topological space). If it's called something else what is it? If Euclid was doing some other kind of geometry than similarity geometry what kind was it?
Vaughan