Most organic molecules feature a lot of hydrogen that essentially serves as a placeholder for all the free bonds of carbon (and there is plenty!), oxygen, and nitrogen. Hydrogen is essentially the default thing to connect to about any organic molecule. And yes, it is primarily taken from water in the grand scheme of things.
To expand on that, hydrogen is just lone protons. Some of those protons pick up an electron, but if it's a proton, it's hydrogen. And considering that nuclear fusion is hard^[citation needed]^, it makes sense that one of the most common things to attach to other atoms would just be the smallest, most abundant, and most simple kind of atom out there.