How does this not become alcoholic?

Short brewing time. Finnish sima is similar - sugar, lemon, water, yeast, ferment for a couple of days. Alcoholic drinks are usually fermented much longer, aged, and processed much more intensively.

It is usually alcoholic. 0.5-2%. Here stores sell most commonly 0.5% one which is regulated to be sold like a non-alcoholic drink.

I understand that Kvass is around 1% alcohol.

By Russian standards, this is "non-alcoholic".

I once got a Root Beer brewing kit as a gift. It noted that a trivial amount of alcohol would be produced by the process (no included or added yeast either!) and it wasn't a big deal.

Usually they remove the alcohol later. The taste is similar to a alcohol-free Guinness (but sweeter).