They are basically the same, but "beside you" usually means you are literally right next to the other person. While "by your side" can also mean this, it is also used in a more poetic sense to mean you support another person or are their friend/romantic partner. So if I write, for example, "My wife was by my side during my years in university," it means she was supporting and helping me, not that she was always just a few centimeters from me! "Beside me" can also be used in this poetic way, but we still generally use it to talk about a close physical distance.

They're pretty much the same and both are plain, common phrases. When used figuratively to mean emotional support, I'd be more likely to say "by your side." There's a third phrase, "side by side" which implies precise physical alignment, almost like marching together. There's a common, but very mysterious idiom, to be "beside yourself," which means to be so emotionally upset and anxious that you can't think clearly or act in a sensible way.

Ye, they are pretty much exactly the same. "Beside someone" may be more common in everyday speech. I ate breakfast by her side. I ate breakfast beside her. She walked beside me all the way to the store. She walked by my side all the way to the store.