Yeah, your right, but if you can think of a better reason why it's worn on the left side let me know. I was told this by my old tae kwon do instructor who is a seventh degree black belt and has been training for over 35 years... He was semi old school. I'm just assuming that the reason why the stripe is worn to the left in both martial arts is the same.
Pretty sure that is for wedding rings, not necessarily BJJ or Taekwondo. The idea of wearing a wedding ring on your left hand because it was supposedly closer to your heart has to do with an old superstition that the vein that feeds your left hand ran straight to the heart and was nicknamed "the vein of love" or something to that effect.
For wedding rings, that seems understandable. Though it may also simply be that most people are right-handed and it makes more sense to wear jewelry on your non-dominant hand. But then there are some cultural variations, and some places have wedding rings on the right hand.
Also, a side preference for donning stuff extends to a lot of things. Medals on military uniforms, handkerchiefs in suit pockets, most shirt chest pockets are situated on the left, you lay the left side of a shirt over the right to button it (for men's shirts anyway), most martial arts uniforms lay left side over the right, etc.
Most of this could reasonably be assumed to do with a preference for righties that eventually just became customary. Easier to grab something in a left shirt chest pocket with your right hand, easier to button a shirt with your right hand, easier to place medals or pins on your left side, etc.
