Does cycling tighten your butt?
Since cycling is primarily based on leg strength, which is derived from all parts of the leg (including the glutes or butt muscles), then yes.

Here's a great graphic that shows some of the muscles used during a typical pedal stroke:
 

And basic anatomy tells us that the muscular system works together so it's virtually impossible to train one muscle exclusively without using others at least minimally.

Now, if you want to specifically target your butt, I'd recommend some supplementary exercises like pelvic tilts/bridges or walking lunges.