The question:
Can someone help me analyze the correctness of this statement, and/or in what environments (or under what modifications) it could be correct?
If $f:E_1\mapsto E_2$ maps circles and straigth lines to circles and straight lines (not necessarily respectively) then it preserves angles, i.e. it is conformal.
What are $E$'s? If I give $E$'s the freedom to be any geometry: plain Euclid's, spherical or hyperbolic, then I believe that the statement is true (or are equivalent at least). Obviously, $E$'s are 2-dimensional and maybe there are ways to generalize, but let's first discuss this case. The statement gets even richer if we allow $E$'s to be some open sets of these geometries, then we get the circle inversion included.
Background:
Not much honestly... I was simply exploring different embeddings of these $3$ geometries into each other (all modeled with a plane and a sphere touching the plane and projections) and was trying to prove that some embeddings (projections) preserve angles, then that they preserve colinearity, then concyclicity and so on... In the process, the importance of circle inversion was lurking and that's where I got this idea. I still have no results in this exploration, just ideas. I just know that the converse of this statement obviously doesn't hold.