I want to have two shapes on top of each other, simplified shown in the image below.
The design should be printed with a hot stamp method, so the manufacturer needs to have the vector graphics of each color in a separate layer. If I simply hide the black layer, the blue circle will display (and print) as a full circle. Is there a way of removing the covered part from the blue circle without having to rasterize the design?
Yes, of course. Go to Windows > Pathfinder then select both circles and click the remove-button in the Pathfinder window.
Ensure that your circles are converted to shapes! Object > Expand
Instead of using the pathfinder option for this, I would suggest you use the knockout group option instead. This will keep all of the shapes involved fully editable.
Below, you can see how I placed one ring above the other and added a stroke to it
Select the group and open the appearance panel (Window > Appearance). Now, click on the underlined word "Opacity" and check the "Knockout Group" option. Yours will not look exactly like this, because I went back to take this screenshot after I finished the process. Your stroke will still be visible.
Here is where we make the stroke invisible. Select the ring with the stroke applied to it and open its appearance panel. Click the underlined word "Stroke" and change its opacity to 0%. Your underlying object should now have the stroke cut out of it and yet everything will still be editable
If your black circle has a white stroke, that black circle will also require a OBJECT > EXPAND APPEARANCE first. And then apply the Pathfinder > Exclude as mentioned by @Malte.
