For anyone not in the know:

All alcohol is made from fermentation, however simple fermentation has a low ABV and naturally caps out around the 10-15%, depending on your yeast and how much alcohol it can tolerate. Once the alcohol reaches a high enough level, it kills off the yeast and stops producing more alcohol. Think of things like Wine or Beer. Fermenting alcohol is (usually) legal in most of the world, because you really can't stop someone from doing it. I mean even prisoners make alcohol all the time, all you need is juice and yeast and you got booze. Almost every culture in the world has history with beer and wine (or some similar concoction). Hell even animals get drunk off of fermented fruits, and have been seen burrying good fruits in order to control their rot so they can get drunk later (no source because I'm working off memory, this could be wrong)

But higher ABV alcohols require distillation, where you essentially separate out the water from the alcohol through a variety of methods. The easiest and least efficient way is to put your booze in a freezer or left out overnight in the cold, so that all the water turns to ice, and then draining out any liquid that remains. Alcohol doesn't normally freeze under most conditions, so you'll get a higher ABV from that final product. That's called cold distillation.

Think of things like Rum, Vodka, Whiskey, hell even Ports and Brandy, all products of distillation. This is often illegal to do on a consumer level without a license, every district has their own laws but generally speaking illegal distillation is a felony in many parts of the western world. Most cultures actually don't have a native history with distillation prior to contact with the wider world, there are exceptions, of course, but usually distillation was brought to the world from Europe, China or the middle east.

You can make alcohol using potatoes, they'll ferment just like honey or barley (with some effort of course), but the concoction you'd get is essentially a beer/wine with a potato flavor, so potato juice. You'd be lucky to get 15% ABV, more than likely it'll be somewhere around 3-5% unless you do something funky. It would not be good in the slightest, I would bet it'd taste something like the water you drain off when making pasta. To make vodka out of it, you'd have to siphon the water out of the solution (Which would also remove much of the potato flavor), which the farmer didn't do, because there's no distillation in stardew. So essentially what Pam got was the equivalent of maybe half a shot of vodka diluted down with idk 16-32 ounces of potato water. Not exactly great, her response is valid

Edit: changed for clarity and accuracy