Why did my mango vodka infusion have a gelatin residue?
It is almost certain that you do not have a gelatin in your mango infusion:  gelatin is formed when an animal connective protein called collagen is hydrated at moderately high temperatures in the presence of water.

Instead, what I suspect happened is that the pectin, a gelling agent present in fruits, including mangoes, has had long enough to dissolve into the liquid phase of your infusion (it will thicken both water and alcohol), making it seem somewhat thicker.

Pectin is the same substance, by the way, that makes fruit jellies and jams thicken&mdash;the same effect has happened to your infusion, just over time, instead of from heat processing.

Assuming you had a sufficiently high ratio of vodka for the alchohol's preservative effect to apply, your infusion should be safe.