The solid state (Peltier) cooler in that fridge probably uses something like 50W electrical power. I would guess maybe a third of that (15W or so) is available as cooling power at normal fridge temperatures, and at steady state (inside of fridge is cold, and stays at the same temperature) it is just enough to balance the heat gain from the outside through the insulation. (Compare that to a mechanical refrigerator with about 3W of cooling power per 1W of electrical power- SEER of 12(*)). So, to use the same cooler to cool a much larger surface area fermentation chamber, you need much thicker insulation, avoid any air or humidity leakage, and you need to take into account the heat produced by the sugar conversion to CO2 and alcohol inside your fermenter. I think 5 gallons of 1.05 wort converted to beer in 3 days produces about 6W (see
How much heat does a fermentation produce (**)).
I'm afraid 15W is just not enough, no matter how good you make your insulation. And you then end up with a bulky container for your fermentation chamber, not very much unlike a small fridge. If you try it, make sure there are fans on both the hot side and the cold side of the cooling element.
All that being said, I built a 270W (total electrical power) solid state cooling system (with liquid-to-liquid-to-liquid-to-air heat transfer) for my fermenters and kegs, since I don't have space for a large enough fridge in our kitchen. It's just very inefficient compared to a normal mechanical refrigerator.
TL;DR: It's certainly possible to build a solid state cooling system. The peltier cooling elements are very inefficient compared to mechanical refrigerators. The problem people usually run into is efficient heat transfer.
(*) SEER = BTU/(W * hr), 1 BTU ~ 1000 W seconds, so 12 SEER = 12,000 W s/(W hr) = 3.3 W cooling power per W electrical power.
(**) this link seems to give an energy output from fermentation a factor two higher than this one:
How to Size a Brewery Chiller | BV Thermal Systems ; or maybe it's just too late in the evening to do the math correctly....