Illustrator :: Eye Dropper Not Picking Up Exact Color?

This has happened during a few different projects I've had:
 
Short Version: I'm just trying to make sure that the colored box prints the exact same color as the PDF's color and it's not.
 
Long Version: Say I'm given a PDF, it's a poster, and it needs to have bleed (a border around it) added. I'll open Illustrator, put the PDF in there, and use a colored box for the additional bleed/border I'm adding. I'll use the eye dropper, click on part of the PDF - it'll give me the color of that spot. I'll take that color and apply it to the colored box, it changes color.
 
The issue is that sometimes, that color the eye dropper picks up doesn't actually match the PDF's true color. I can fill in the colored box with whatever color the eye dropper picks up, but it's often a weird color, like with CMYK number values being at 98.35% or something And even when the color value on the PDF and box are the same, you can see a slight difference between that colored box and the real PDF.
 
I'll even export that to a PDF and make sure it's correct, using the Object Data Tool or whatever it is, and then using the eye dropper there, and seeing that the color for my box and the PDF are at the same CMYK color values. And you can still see it's incorrect very slightly, and then when you print it it's obviously not the same. Even when I try picking it up in HSB or RGB, it's no good.