Is there a name for the class of transformations on the Euclidean plane (or projective plane) that preserves lines?
They are not all affine transformations; consider a perspective projection $p$ in 3D Euclidean space (center of projection at $(1,1,0)$) from the $xz$-plane to the $yz$-plane. This map is not affine.
Could it be that all transformations on the plane that preserve lines are results of such projections?
A transformation of a projective plane which preserves lines is called a collineation. In every projective plane, a projective transformation will be a collineation. In the common Euclidean plane with the real numbers as the underlying field, the converse can be shown as well: every collineation is a projective transformation. If the underlying field has a non-trivial automorphism, the way the complex numbers do, this is no longer true. For example, one can compute the complex conjugate of every coordinate, thus preserving lines although that operation is not a projective transformation.
Source: Lectures by the author of Perspectives on Projective Geometry.