DNA is an acid. How does it behave like one?

I don’t think so. Gelatin is just denatured collagen, I believe. I don’t think it has any “muco” component (which just means that polymers of sugar molecules are attached to the protein). Collagen is the main structural protein in vertebrates. Cartilage consists of it, and there is a lot of it in bone. When you heat it up in water in the right way and the protein chains unwind, then get tangled up with one another and so set into a jelly.

Mucoproteins are generally in things nature intended to be gooey - like, uh, mucus - not things that cooks make gooey.

In biological systems, the pH (Proton concentration, acidity) is regulated, and the acids and bases adjust their protonation state accordingly: Acids are converted to their conjugated bases by giving up a proton if the pH gets near or above their pKa (negative logarithm of the protonation equilibrium constant), bases are converted to their conjugated acids if the pH gets near or below their pKa. Since the backbone phosphoric acid/phosphate groups of DNA are strong acids, they are deprotonated (connjugated base) at physiological pH.