How to convert png to stl?
I suggest your objective can best be accomplished by converting the image to a single color vector file. You can do this with Inkscape (free, Linux, Windows, Mac) by combining the built-in bitmap tracing feature with some manual editing. I attempted to do so, but the coarseness of the image would result in excessive manual edit time. It may even be faster to reduce the opacity of the image and create a manual tracing on a second layer.

Once you have a vector file, you can use any number of 3D modeling programs to convert the file to a 3D STL. Fusion 360, Tinkercad and OpenSCAD allow import of SVG files to be extruded to user-specified thickness.