Can you tighten your stomach muscles while riding a bicycle?

Not a lot, bike riding is mainly about leg and and a little about back muscles.

If you look at a dedicated road bike or cross-country MTB rider you see them riding hunched way over forward.

Part of that is aerodynamics, but mainly it is because the upper body acts as a counterweight and provides an opposing force for the legs pushing down on the pedals. And it's that lean forward that puts a strain on the back muscles.

Riding out of the saddle, as MTBers do on difficult trails, or road riders when climbing hard will get your upper body more involved.