The toilet and the stomach bacteria rarely encounter one another, unless you throw up on a regular basis.

You're probably referring to gut bacteria, which live in your colon. It's true that a lot of research is being done towards these bacteria, and they appear to play an important role in various systems.

However, stools are usually coated in a thin layer of mucus -- which makes them a bit shiny and serves as a lubricant. It's unlikely that enough stool interacts with the toilet bowl to get much interesting data out of it. And even if it did, it would probably be contaminated by the presence of other people's stool.