That’s a giant no. All of the stuff that makes a fart smell like a fart are too volatile to store.
Yeah, the main constituents are stable enough, but methane alone does not a fart make. Besides, not all farts contain methane.
The stuff that smells is what matters for freshness. Hydrogen sulfide (rotten egg), methanethiole (cabbage-like), scatole & indole (poop smell), dimethyl sulfide (garlicky) are the ones that are less than pleasant.
But there’s stuff like lemonine and pinene as well. They don’t smell unpleasant to most people, but in the wrong proportions, they can contribute to unpleasantness alongside others.
And all that’s just the main, common ones. You get traces of stuff like cadaverine sometimes.
The thing most (actually all, but I want to give leeway for the internet) of that have in common isn that they react with other things to some degree or another. They interact with each other in an enclosed space. Hydrogen sulfide is (iirc), the most stable of them, but it isn’t exactly going to sit unchanged in a container forever with the other ones.
There’s actually a decent amount of research into the digestive processes that involve gasses because they’re a big indicator of how things are working in the gut. There’s patterns of flatulence contents that vary between people with various digestive issues (like IBS, and IBD in terms of chronic conditions). Active infections change the patterns during infection, and may cause long term changes as well.
An interesting side note is that the chemicals that make farts is that they’re also found in rotting bodies, and rotting vegetation, though the proportions and exact chemicals vary between all of those. Digestion is controlled decay, if you want a pithy little phrase to piss off pedants :)
It isn’t even an inaccurate phrase; a lot of what happens in decomposition of animals (including humans) is driven by enzymes and bacteria, including the same ones found in our gut. But it’ll piss off pedants anyway, because it isn’t exactly the same thing.
There’s a reason that feces, flatulence, rot, bad breath, and even burning things can share smells in common. There’s a reason skunk spray, or musk, or even stale sweat have similarities that our noses can detect. Chemistry, chemical reactions.
They’re also partially done by itty bitty critters crawling on and in everything. Those smells in our farts, poop, and rotting flesh are all germ farts. They’re the waste products of bacteria (and fungi) eating our waste, the waste of everything. Those microbes are chemical factories.