That means I shouldn't worry about my shortness of breath while climbing mountains.

That depends. Do you also face breathlessness while doing routine tasks at home? If yes, you should be worried and you need to hit the gym.

Do you only feel breathlessness while hiking? If yes, that's pretty normal for everyone but you still need to hit the gym. Hitting the gym is never a bad decision xD

The best way to train lungs is to jog at a steady pace. Also, doing breathing exercises. Breath in to the maximum capacity of your lungs. Hold the air inside and then exhale out in 4-5 steps. Do this atleast twice for two minutes in a day. Alongside breathing issues it will also solve your anxiety.

Hmm... If you experienced it on the highway that means you were probably in a vehicle and your body didn't exert (no climbing), right? And you still went through hypoxia?

No I meant when I got out of the bus at the Pak-China border. Still, there was no sort of hiking or Trekking, only walking yet I still felt shortness of breath (pretty normal at that altitude)