Does a simple photograph contain more information than a complex painting?
It depends how you define the term "information" or "entropy".

The conventional definition of entropy of an image is to think the image as a two-dimensional matrix of pixels and 
$$H = - \sum_k p_k \log_2(p_k),$$
where $p_k$ is the probability, which is calculated from histogram, associated with gray level $k$.

This kind of entropy is correct if we ignore the correlation between pixels. For example the two images have the same entropy by this definition.

 

It is not true if the correlation between pixels is considered. For example if the color first pixel in top-left has probability $p_k$, the next pixel surely has the same color and its color does not have the same probability $p_k$.

We human being, with you as an example, use this kind of correlation to perceive the images. This kind of correlation are called "details" and, therefore, we/you expect that rich-detail images should have more information/entropy than simple ones. This is the reason why you have found it counter-intuitive.

PS:

I tried calculating the entropy of the two images you have posted, but they are not different "by one or two orders of magnitude" !!!! 

"Along the River During the Qingming Festival" entropy about 7

"The sky" entropy about 6

They must not be the same files of the professor.