In all the examples of categorization with neural networks that I have seen, they all have training data that has one category as the predominant category or the label for each input data.
Can you feed training data that has more than one label. Eg: a picture with a "cat" and a "mouse".
I understand (maybe wrong) that if you use softmax for probability/prediction at the output layer, it tends to try and select one (maximize discerning power). I'm guessing this would hurt/prevent learning and predicting multiple labels with input data.
Is there any approach/architecture of NN where there are multiple labels in training data and multiple outputs predictions are made ? or is that already the case and I missed some vital understanding. Please clarify.
Most examples have one class per input, so no you haven't missed anything. It is however possible to do multi-class classification, which is sometimes called joint classification in the literature.
The naive implementation you suggested with a softmax will struggle as the outputs on the final layer have to add up to 1, so the more classes you have the harder it is to figure out what the network is trying to say.
You can change the architecture to achieve what you want however. For each class you could have a binary softmax classifier which branches off from the penultimate layer or you can use a sigmoid, which doesn't have to add up to one (even though each neuron outputs between 0 and 1). Note using a sigmoid might make training more difficult.
Alternatively you could train multiple networks for each class and then combine them into one classification system at the end. It depends on how complex your envisioned task is.