Would our digestion function any differently if we secreted something else, like sulfuric or nitric acid, instead? I'd assume an acidic environment may be required, but not sure if chloride is also required as well!
Perhaps we just use it out of convenience—e.g., presumably blood serum $$\left[\ce{SO4^2-}\right] + \left[\ce{NO3-}\right] \ll \left[\ce{Cl-}\right]$$ and channeling polyatomic ions across the membrane may be significantly more difficult than doing the same for chloride as well!
*Edit: In fact, Wikipedia states
In the stomach, chief cells release pepsinogen. This zymogen is activated by hydrochloric acid (HCl), which is released from parietal cells in the stomach lining. The hormone gastrin and the vagus nerve trigger the release of both pepsinogen and HCl from the stomach lining when food is ingested. Hydrochloric acid creates an acidic environment, which allows pepsinogen to unfold and cleave itself in an autocatalytic fashion, thereby generating pepsin (the active form). Pepsin cleaves the 44 amino acids from pepsinogen to create more pepsin.
So, while maybe something like $\ce{H2SO4}$ or $\ce{HNO3}$ could work too, hydrochloric acid is just more convenient to use?