How to count categorization instances in a NN?
But your intuition about regression is right. Basically, you have multi target regression problem. Each category is a separate target. If your get float output as prediction, you can round it. Float output also makes sense, because it shows you how confident is your model about prediction and the expectation of the number of bought items.
Number of bought items is structured as ordinals, so it makes sense to predict it as multi target regression. Moreover, you may not know the upper limit of the number of bought items, so it makes no sense to predict number of items as categories.
Look at this for reference:

Neural Network for Multiple Output Regression
Resources for learning about multiple-target techniques?