Edited to address the edit:
How can I sympathize and support my brother about X while supporting with my dad in him trying to reprimand my brother about throwing a fit about a "newly received gift" without them both being angry at me?
The way I see it, your brother didn't throw a fit about a "newly received gift". If your brother didn't know what he was getting and acted the way he did, it would be as you say. But it's much more complicated than that. Your brother threw a fit because, once again, his express wishes were ignored. That's very, very different. It's not a fine line. It's a big red flag for your dad.
My answer doesn't change otherwise. It's not your job to support your dad's opinions, especially if they entail the hurt of someone you love. It's his job to shape the values of his children. Say he had a fight with your mom. Would he be right if he demanded you support him in his actions and was hurt when you (rightly) refused to get involved?
In love, you console and support what is supportable for both.
If you need a more specific answer (a sample dialog), leave a comment and I'll edit further.
As @baldPrussian said, please frame it differently in your mind than "taking sides". Taking sides implies being against one party. There's a difference between being against someone and being disappointed in something they've done.
When I read your question, I was disappointed in your father's choice. A better course to take, since the present wasn't going to be a surprise anyway, would have been to ask your brother if "this new version" would be even more desirable than what he asked for. But he didn't do that. You say he has often disappointed your brother in a similar fashion before. That's unfortunate, and it hurts your brother.
Your brother was deeply disappointed. I don't blame him. I don't even condemn the way he reacted if this is something that happens often. How is a depressed 13 year old who has placed in your parents' hands some hope for feeling good about something only to have it dashed supposed to act? He could have handled it better, but he didn't, and that probably hurt your father.
It's sad for both of them.
How can I sympathize and side with my brother about X while siding with my dad in him trying to reprimand my brother about throwing a fit about a "newly received gift"?
How about changing the scenario to:
How can I be supportive of both my father and my brother?
Be consoling, not condemning. Console your brother because you understand better than most how sad he is. Console your dad in that he thought he was doing the right thing. But if your dad demands of you to agree with his criticism of your brother, tell him that's not your job.
It's the job of the parents to help guide the child.
I hope your family is involved in family therapy. If they aren't, you might suggest it, or talk to your school guidance counselor to seek a way to advance this issue. I wish you all the best.