With the help of the compositor I added glare to my object and managed to render it with alpha. But is there a way to separate the glare that was added on top?
In the end I want to have two images. One with the object and one with the glare.
If I understand you correctly, you want to add a glare to the brights parts of your image. You can filter out these parts with RGB curves and then add your Glare. After that, you can alpha over against black the result. Each of your two layers can be Saved with a File Output node on your hard disk. To Combine them simply add the glare layer over your original.
EDIT: As mentioned below you dont need the AlphaOver if you set the Mix value of the Glare Node to 1.
If you move the Mix slider in the glare node to 1.000 it will only show the glare on a black background, you can then save the image.