Shannon's entropy for an image
Shannon's entropy works as a hashing or fingerprinting function here; they are not injective which means we have to be careful not to interpret too much into relations of values.

If $H(I_1) \neq H(I_2)$, then images $I_1$ and $I_2$ are certainly not the same.

If $H(I_1) = H(I_2)$ (or even $H(I_1) \approx H(I_2)$), however, we know nothing. The images might be similar, but the values can also be close because of non-injectivity.

We would like to have some smoothness, that is that two images are more different if $|H(I_1) - H(I_2)|$ is larger. Whether this is the case is certainly a quality criterion of such fingerprinting functions. Given your example, it does not seem as if entropy was a good fingerprint in this sense.

It would be confusing if any kind of entropy would be. Any rectangle of one color holds no information (entropy zero), yet two rectangles of different colors are as different as two images get.