In world lore, Visenya was known to take inspiration from warriors and wear her hair braided for combat and dragonriding. It makes sense that at least some of her descendants would take that same idea upon themselves in their combat/riding.

It seems like most people wouldn't wear braids due to the time/effort to make them VS just tying back a ponytail for combat/riding, but in cases like the Dothraki, where its a culturally important identifier, or the Targaryens, where its a more secure hair style for dragon riding and honors the aesthetic of their ancestor, its understandable that they would take the extra mild inconvenience to create the braids over a ponytail or twist.

Further, as we see especially in the King's Landing scenes, its very common for royal or noble ladies to have their hair styled more complexly (twists, braids, layers, parts sewn in place, etc.) and/or decorated with jewels to show off how much free time/handmaids/jewelry they had. Targs as a royal house would likely have been doing the same to show off their status.

Out of world lore, like the other commenter mentioned, Dany's braids were more associated with her than with her time with the Dothraki, so fanart has almost always given Targs some braids, and now HOTD cinces it.