When I apply a color with the exact same RGB as the fill for 1 shape and the stroke of another, the shape with the stroke shows a slightly lighter color. Why does this happen and what can I do to correct it?
There's a good chance that the effect is caused by your custom stroke. If the stroke shape isn't completely black (because, for example, it was made using CMYK black insted of RGB black), it will show as a slightly lighter variant of your chosen stroke colour.
To solve your problem: expand the shape (Object > Expand; or Object > Expand Appearance...) and give it the correct colour as a fill colour.
A more desirable approach than expanding the shape is to edit the brush colorization method. If you change it from just "tints" to "tints and shades" it will ignore the shade of CMYK black used in the original brush. Be sure to click "apply to strokes" when it asks you as you exit the window in order to make the change retroactive.
I had the same problem. I think I've found two possible solutions. Since nothing I did would change the color to make it 100% black, I tried these:
cmd-c, and 3-4 times cmd-f (This is the lazy solution, but it kind of triple/quadruples the lines.)
Drag the brush onto the artboard. Overlap a black rectangle and do a pathfinder (make sure to delete the parts not needed, it might be tedious depending on the brush), apply black 100% and this time around the shape is 100% black. Then all you need to do is drag the brush back into the brush window and make sure the settings are like the original. Then you apply the new brush.