Replace a 2.0" tyre with a 2.1" studded one? I'm looking at replacing the tires on my GT Aggressor commuter bike with studded tires for the winter.

Based on the current tire sidewalls (the rims themselves don't actually have the width or diameter of the wheels on them), I'm looking at getting a 26x2.0" studded tire set.

Thing is, a local winter bike gear supplier carries one pair of 26x2.0" studded tires, and upwards of a dozen of (generally higher-quality, better-reviewed, knobbier) 26x2.1" tires. Would putting a 2.1"-wide tire on a (supposedly) 2.0"-wide rim be dangerous for the bike or the rider?
Look at what clearance between the tire and the frame do you have now. Often the tightest place is the bottom stays.

Changing 2.0" to 2.1" increases the tire diameter by about 2.5mm ("about", because the rim width stays the same and does not proportionally increase), i.e. the tire grows ~2.5mm upwards and ~1.3mm on each side. Not all tires are measured equally, some "2.1" tires may be wider than others, so add one millimeter more to be on the safe side. Besides that, knobs make the tire higher, attempt to account for that too.

If you conclude that there is enough space to fit the wider and knobbier tire, there are no other obstacles to replacement.

(Not directly related to the question, but I'd strongly recommend buying higher-quality studded tires. This is an item which brings much more value for the money than any other bike part.)