The container for the wood filler noted it was stainable.
Yes but it did not tell you how much stain it would absorb. Not did it specify what kind of stain.
the filler did not take the stain at all.
No, and it is not going to.
I am certainly not an expert but I have done a bit of wood working. Welcome to the frustrating world of wood staining and finishing.
There is a number of types of wood stain. They break down into two major types, wiping stains and penetrating stains. The wood patching material may absorb some penetrating stain but is unlikely to absorb as much stain as the surrounding wood. It will not absorb any of a wiping stain. It is a wood patching material, and not a wood filler. Wood filler is another material that fills in between the fibers of certain open grain woods.
You didn't specify but you are probably using a wiping stain. A wiping stain is pigments of stain suspended in a carrier, usually something like mineral spirits. You apply the stain, let it dry for a short period then wipe off the remaining carrier and pigments.
If you let a can of stain sit for a while, then pour off the liquid you will see that the pigments in the bottom are of different sizes. Usually the darker pigments are larger in size. The stain does not actually color the wood fibers. If it does, it is only slightly. When the carrier evaporates the dried pigments are trapped in between the fibers and in the groves left from sanding. That is why you can remove almost all of the satin by re-wetting the wood with more stain and then immediately wiping.
Sanding makes a difference. Sand to 100 grit and you will get a certain amount of staining. Sand to 120 grit and you will get a lighter stain. Sand to 600 grit and you will get almost no stain. The courser the sand paper the larger the scratches left, the more space for the stain pigments.
The wood patching material has no grain. A penetrating stain might actually color it a little but not as much as the surrounding wood. It is also harder than the wood so the groves left by the sanding will be smaller and there will be less room for the stain pigments to reside.
Depending on the color you might try a wood dye (even ink from a felt tip marker) to get the patching to something close with top coat of stain. However, most likely you will not be satisfied with the results.
My suggestion would be to remove the patching with a drill bit then remove the screws. Counter bore the screw holes. There is a difference in counter sinking and counter boring. Reinstall the screws and plug the holes with plugs. Sand flush and re-stain.
You can purchase plugs but for best results they need of the same kind of wood as the wood you are plugging. You can purchase a set of plug cutters and make your own from the same material you are using. If you used plywood for your end pieces of the book shelf you will have cut them from the same kind of wood the plywood is made of.