I am looking for a mathematical way to check if the distribution of points inside some region (almost never a proper form) are evenly and uniformly distributed through it. Do you think this is possible? 
The left one is a well distributed example, and the right one is not evenly distributed. Might be a bad example regarding the shape, but the point is that it is never a circle, rectangle etc. I tried a couple of concepts, of which one was to divide the region in 100 small squares and check each block separately but because it might be very oddly structured it did not work well with small and/or extended parts (like the 'tail' on the upper images).
Thanks in advance!
The energy distance[1] and the kernel two-sample test[2] provide two-sample tests for arbitrary distributions, defined using samples.
In your case, you might be able to arbitrarily create sample sets that are uniformly distributed, then measure the distributional distance (by energy distance or Maximum Mean Discrepancy [MMD]) between candidate sets and uniform sets.
When these measures are larger, the sample is farther from uniform. If you're trying to find a decision threshold to reject the hypothesis that a sample set is uniformly distributed, those papers also provide test statistics.
[1] https://en.wikipedia.org/wiki/Energy_distance
[2] http://www.jmlr.org/papers/volume13/gretton12a/gretton12a.pdf