There is a published paper ("On 2-Reptiles in the Plane") proving that there are exactly 6 rep-2-tiles. But this restricts the definition of a rep-tile to a dissection into directly similar parts. If you allow inversely similar (reflected) parts there are more. These include an infinite number of different parallelograms, parameterised by the value of the smaller angle. If you change the question from the number of rep-2-tiles to the number of classes of rep-2-tiles interconvertible by affine transformations there are at least 10, and possibly 12 (I haven't succeeded in demonstrating that two of them tile the plane).
If you relax the definition to allow irreptiles (dissections in which the parts are similar, but of different sizes) there's another infinite class - of right angled triangles. And there are at least another 8 irreptiles corresponding to the polynomials x+x^2=1 (the golden bee shown above), x+x^3=1 (4 tiles) and x+x^5=1 (3 tiles).