Is rigibody velocity frame rate independent?

Your physics shouldn't have 1000 fps, but something around 50, meaning a fixed timestep of 0.02. That's the default value and changing it shouldn't happen without a good reason.

Now, Rigidbody.velocity is just some value that you can change. Nothing really happens when you change it. But when, in the next fixed update step, the physics engine does its work, it will move the all rigidbodies according to their velocities. Since it does that in a pseudo-constant frequency, changing fps should definitely have any influence on physics calculations. And chaning the velocity, at what time whatsoever, can't influence that.

As @misher said, you can watch input wherever you want and still apply it in FixedUpdate. You just need a variable that serves as a flag:

 private bool jumped;

 private void FixedUpdate()
 {
   if(jumped)
   {
     Jump();
     jumped = false;
   }
 }

However, as I explained, the velocity property already serves as this... "container" for changes to be applied in the next physics update.

So, as to why your velocity varies on mobile... do you use Time.deltaTime somewhere? What other code influenced time, phyiscs in general, or this rigidbody?