If I understand the U-Net paper correctly, the NN output is segmentation of known objects on the image from the background. In other words, the network will try to mark all the pixels which are part of the detected objects, but it won't distinguish between object instances. Is my understanding correct?

If yes, is there an approach which would allow me to differentiate between object instances in the image?

For example, if we have a photo of a street I would like to know not only which pixels are taken by cars, but also "enumerate" the said instances (even partly overlapping ones).

The reason I am asking is that I believe NN already learns what the object is and should therefore intrinsically "detect" the instances, however the U-Net input and output do not match this expectation directly (or I have misunderstood something - quite possible as I am still a newbie at ML).