Probably comes down to their diets, which are (very broadly) similar but very different in the amount digested and how long it spends in the GI tract. The big vegetarian dinosaurs appear to have gotten a fair bit of their food from treetop grazing, which is pretty different from the grasses, shrubs and water plants geese live on in the wild. Plants have evolved quite a lot since the days of the dinosaurs, and geese have evolved with them, so it's hard to speculate meaningfully as a hobbyist. Probably some actual paleontologist who could say with more confidence.
As an aside, there have been a number of D&D cartoons over the years dealing with the obvious dangers of something the size of a dragon soaring overhead and voiding itself (intentionally or otherwise) on targets below. One might reasonably assume that the draconic "elemental" affinities shown in their breath weapons and immunities/resistances are echoed in their excreta as well. A passing black dragon might be the very worst type of acid rain, red dragons a menace akin to an incendiary bombing run, greens a chemical weapon attack. The prospect of whites leaving hailstones in their wake is a bit more absurd but (as anyone who's passed a kidney stone can testify) might explain why they're so cranky. Who knows what's up with blues - does a stream of their urine conduct electricity like a inverse of the electrical fence adage?
I've heard it from deer hunting enthusiasts, albeit rarely. Certainly possible that the scouts or 4H might teach it. The terms are archaic but technical language tends to preserve specialist terms well beyond general linguistic shifts, and hunting parlance is definitely pretty technical at times. And that's the thing, I don't think fewmets/fewmishings was ever just another general word for...um, poop? The filter allows poop, right? They've always meant something pretty specific as far as I can tell.
