I want to make a scene where there are a lot of 2D sprites /textures that look like they are cut ou of a piece of cardboard. There must be an easier way then to model this, right?
So here’s an example of the sprite with absolutely no thickness:
You can’t create any kind of thickness from alpha textures. Even if you add a Solidify modifier to the mesh, you will still not see a “rim”, but just the front and backside.
What you could do is to vectorize the bitmap to an outline in e. g. Inkscape, import that into Blender as curve, extrude with the curve tools and work from there. This will at least spare you the effort of having to manually model the cutout shape: