Does store-bought Kombucha spoil? If so, how can I tell?
When they say "probiotic", what they mean is that there are microbe cultures present in the kombucha.  It's really nothing more than a sweetened tea which has been fermented by a symbiotic mix of yeast and bacteria.  The odor and distinctive flavor is the result of a low alcohol content and acetic acid (the same acid found in vinegar) produced by the fermentation process.

While these cultures do tend to edge out other bacteria that would spoil the drink and potentially cause illness, it can eventually go bad like anything else.  More to the point, because the cultures are still alive, they'll continue the fermentation process as long as they can, especially once you open the container and start introducing oxygen.  This can throw off the balance of the culture, but it can also just make the kombucha unpleasantly acidic.  Once it gets to that point, it might still be effective from a probiotic stance, but the acid could have an adverse effect on your digestion.

If you keep it sealed and refrigerated, it'll last a couple weeks, but a week or so is probably the effective maximum when you're opening the container frequently. You'll want to stick to purchasing an amount that you can drink within a week or so, but don't worry overmuch if it has a slightly vinegary tang and odor.