In the "This die" rule clarification the question is asked:

Question: I activate a Clone Trooper (Legacies, 38) with a Training (Spirit of Rebellion, 125) on it. Can I reroll both of its dice?

Answer: No. You can only reroll one Clone Trooper die of your choice. “This die” always refers to a single die, not multiple dice.

How does this ruling affect Palp & Aurra Sing's character abilities?

It doesn't really. See, you resolve all dice individually(even if it's during the same action) unless their modifying or modified die. So Aurra and Palp both get to do what they want as long as they have multiple die to resolve.

The issue with Clone Troopers effect is that the activation is a singular event, you only activate once outside of things like fient & other readying effects. Since you're only activating once, you get only one trigger of it's effect. Since the effect only refers to "this die" which is singular, they only get to reroll a single die instead of both. If the effect was "After this die is rolled into the pool" it would have worked on both.