You can always measure the specific gravity of any liquid with a hydrometer, but unless you know the starting gravity (as in, pre-fermentation), the reading won't actually tell you much. This on top of the fact that the byproducts of a mixed fermentation (alcohol, acetic or lactic acid, &c) will all have different densities makes a hydrometer reading less-than-reliable for telling you how much sugar is left, though it can probably get you in the ballpark. Of course it's fairly easy with beer (as long as you know the starting gravity) because the only significant byproduct is alcohol and you can easily estimate, by comparing to the original reading, how much gravity is left based on the density of alcohol.
If you were to get in the habit of taking gravities regularly with your kombucha, you could probably figure out when a good time to bottle is. This will depend on having a decent idea of how low the gravity will go, which can be tricky with mixed/wild cultures. Also, though I've never taken gravities on my kombucha, I imagine if you like it dry/tart it might go below a specific gravity of 1.000/0° Plato, which means you couldn't measure it with a regular hydrometer.
The good thing is that kombucha ferments much more slowly than typical cultured-yeast ferments will, so you're less likely to over-carbonate. A good, no-frills method is to bottle some up in a plastic soda bottle and use the feel of squeezing the bottle to judge how much pressure has built up, which will give you a good idea of carbonation levels.