Simple answer: because the human body has chloride ions available, and it doesn't have nitrate, sulfate, bromide, iodide, or other anions available.

Longer answer: when a strong acid is dissolved in water, it really doesn't matter what kind of acid it is, because nearly all of the protons dissociate from the anions. This means that, in stomach acid, what you have is a lot of chloride ions floating around in solution, and a number of protons in solution. Were you to dry your stomach contents out, then the protons would join back up with the chloride to become hydrogen chloride gas, but in your stomach, the protons don't really care that there are chloride around and vice-versa. If you were to mix nitric acid into stomach acid as well, then you would just have a bunch of nitrate ions, and any individual proton won't remember whether it was part of a hydrogen chloride molecule or a hydrogen nitrate molecule before being dissolved. For that matter, the chloride ions don't know whether they used to be part of hydrogen chloride, or sodium chloride, or potassium chloride, and you have significant amounts of both sodium and potassium in your body as well.

Everything is a product of evolution. Chloride ions, Cl-, are simple ions that are quite common on Earth, and they have only 1 atom each. HNO₃ would be a problem, because nitrogen gas, N₂, is very stable - it has 3 bonds between the 2 nitrogens. Plants and animals can't even convert nitrogen gas to nitrates and ammonia. This is done by special nitrogen-fixing bacteria that live on root nodules of plants like bean plants and other legumes.

Here you have an interesting link: "Pepsin Pearls
Parietal cells within the stomach lining secrete hydrochloric acid that lowers the pH of the stomach. A low pH (1.5 to 2) activates pepsin. Acetylcholine, gastrin, and histamine stimulate the proton pump in parietal cells to release hydrogen ions and decrease pH." May 9, 2021