Images do not tell me a lot, but I can see it has two arms. I have no idea about this object, and Turbosquid has a few skeletons on its site. In other words, I don't know how it is created.
There are two main options; why do you see two arms after rotating? The first one, the Command/Control key, was down (and has created a copy, or the setup has two skeletons and switches between the two to provide IK/FK options while avoiding a Joint based setup. Both are guesses, and nothing that I like to do here.
So, go to the Object Manager> Object> Project Information, and check how many objects are in the scene. Close it. Rotate the other arm. Get the Project information again and check if there is now one more object.
This will answer if you have a new copy or the setup is done this way. (While both are visible based on a setup, that would lead to problems later on. Again, these are just guesses. I would need to see it, but you can post it in public. You could send it to [email protected], and I have a look at it. Please note, I do not move the discussion to email; that needs to stay here, so many people hopefully benefit from it.
Animation can be done as Forward- or Inverse- Kinematic. Please have a look here
You could also check out the Content Browser> Skeleton.
There you pick the Spline-based controllers, click and rotate or, in some cases, move them. Those Controllers get keyframed.
All the best