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We know that Harry and Voldemort have the same wand core from the tail a phoenix, and in fact the exact same phoenix, Fawkes.
We know that wands have allegiance, and that it can take time to master a wand.
My questions is given they have the same core (and potentially the same core attributes), could Voldemort and Harry have won each other's wand's allegiance?
If so would they have been equally as gifted with each other wands?
I think you're misunderstanding how winning wand loyalty works in Harry Potter. Wand cores aren't the crucial point. What matters are the circumstances in which one wins a wand from its original owner.
“I took this wand from Draco Malfoy by force,” said Harry. “Can I use it safely?”
“I think so. Subtle laws govern wand ownership, but the conquered wand will usually bend its will to its new master.”
“So I should use this one?” said Ron, pulling Wormtail’s wand out of his pocket and handing it to Ollivander.
“Chestnut and dragon heartstring. Nine-and-a-quarter inches. Brittle. I was forced to make this shortly after my kidnapping, for Peter Pettigrew. Yes, if you won it, it is more likely to do your bidding, and do it well, than another wand.”
“And this holds true for all wands, does it?” asked Harry.
“I think so,” replied Ollivander...
(Deathly Hallows, Chapter 24, The Wandmaker).
You can win a wand's loyalty by disarming the person concerned, as Draco Malfoy did to Dumbledore. You can win its loyalty by physically snatching it out of someone's hand, as Harry did to Draco. You can win its loyalty by killing the previous owner, as possessors of the Elder Wand did down the centuries. Whatever method you employ, you have to conquer the person who currently has the loyalty of that wand.
At this point, you hit a snag. You have to conquer Voldemort, either by disarming him (unlikely), killing him (impossible with his Horcruxes intact) or beating him in a fist fight (possible, I suppose, but he'd probably Avada Kedavra you long before you got near enough to fight him). Even when Voldemort started using the Elder Wand he still had the loyalty of his former wand. Harry would've had to defeat Voldemort in open combat to win the loyalty of that wand. Such an open confrontation with Voldemort was never sought by Harry until the very end of Deathly Hallows. Most of the time Harry was focused on escaping Voldemort, not fighting him.
If Harry had won Voldemort's original wand would it have worked better for him than it would've for another wizard? Probably. After all, the wand chooses the wizard.
“You talk about wands like they’ve got feelings,” said Harry, “like they can think for themselves.”
“The wand chooses the wizard,” said Ollivander. “That much has always been clear to those of us who have studied wandlore.”
(Deathly Hallows, Chapter 24, The Wandmaker).
Harry's wand chose him, partly because of the connection of the cores. It makes sense that it would've responded favourably to Harry, given that its brother was loyal to Harry. However, the logistical challenges involved in winning Voldemort's wand in battle prevented Harry from ever finding out.
