This is a hidden feature in Illustrator which I did not find in the documentation.
In Photoshop the Eyedropper tool is used across documents without any interaction of the user. You can pick colors of an opened document in the background to use them in your current document.
In Illutrator it does not work that way. But there is a way to pick colors from other opened documents which is quick and helpful.
When you use the Eyedropper tool in Illustrator with a single click, you are picking the color and other attributes of objects to reuse them or apply to the current selection. But if you keep the mouse while dragging the Eyedropper will work outside the document and take colors from the outside world, even from the colors in the background.
Imagine you have two documents. The current document on the left with a selected purple rectangle.
If you drag the Eyedropper until you reach the green rectangle on the right side, you’ll apply the same color to the selection. Be aware that if you go over the border of the rectangle you will pick the color displayed instead of the stroke color.
This is the result of the action. I tested this behavior in Illustrator CC 2014 and Illustrator CS6 and it works consistently, so it could be an old feature.
Please leave your suggestions in the comments.