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.