This may not seem trivial but yes, the models we train can potentially learn a variety of things they weren't intended to learn. There are already some examples in computer vision. A typical convolutional network learns things like edge detection, various potentially useful masks etc. in the early layers while learns more high-level features like eyes, nose etc. in higher layers.
It is reasonable too. Given the dataset size is moderately high and the model is trained for long enough, a sufficiently deep network learns various kinds of hidden representations, which may not even be specific to the task at hand. This is the reason transfer learning works very well even on a host of different datasets.
This is limited since not all the learnable things can be described using mathematics. So, the answer is a surprising no. The model does learn some extra things other than the task at hand.
P.S.: There was also a case when a group of researchers trained a model to make a robot walk. It turned out the robot had learned to recognize faces too and reacted in different ways on seeing different faces. I saw the video on YouTube a while ago and couldn't find the exact video to post the link here, anyways.