Mushroom is another name for the culture. It looks a bit like a mushroom.
Sugar will feed your candida. It all comes down to how much sugar is in the finished drink.
I wouldn’t drink it. They may load it full of sugar (and probably do to make it taste nice). If you make your own, however, you can make sure that the finshed product is very low sugar.
Kombucha with very low sugar will taste almost like vinegar. If it tastes “sweet”, or not much like vinegar, it will feed your candida.