When you use a camera to “capture” a picture of something, lots of languages describe the situation using the verb “make” (e.g. ein Foto machen in German). English speakers, however, use a different verb.
The process of pointing a camera at an object and pressing the shutter button is called “taking a picture” or “taking a photo(graph)”, for example:
The reason why “making a photo” does not sound natural is that “making” implies “creating”, and when you take a photo, you do not really create it; you merely capture an existing scene. The distinction becomes more apparent when the word “picture” is used instead of “photo” or “photograph” (using the word “picture” is also much more common in everyday conversation):
I made a picture of my brother. = I drew a picture of him.
Since making implies creating something new, when you say you “made a picture”, people will likely assume that you drew or perhaps printed it, not that you just took a photo with your camera. Note, however, that professional photographers sometimes speak about “making a photo” when they are the true creators of it, i.e. when they prepare the scene, set up the lights, retouch the picture, etc., to make the final product.