Nope, you can do it on a hybrid. In fact, I think it would be easier to do on a hybrid because of the fact that it's heavier and less responsive/squirrelly than a road bike. (I could ride no-handed on the bike I had as a kid, but not on my roadie!) I'd imagine it would be easier (for me, anyway) to reach the bottle on the down tube rather than the seat tube, since you're reaching down in front of you, rather than down and under you.

I'm still trying to master this, but for what it's worth: I think it's a mixture of core strength, balance and practice. I think there's a number of threads floating around on the forum about it, so try a search. One of the tips I seem to remember seeing is to break it into individual steps and visualize each step.