Welcome to Harry Potter Theory. Today we’re discussing Harry Potter, Dumbledore’s Phoenix- Fawkes, wandlore, and Lord Voldemort. More specifically, we’re going to be addressing a couple of things.
1. WHY Fawkes’ Phoenix Feather CHOSE Voldemort
&
2. WHY Fawkes’ Phoenix Feather Chose Harry. We know that Fawkes’ supplied the tail feather for both Harry AND Voldemort’s wands, and I’ve got a sinister theory for why.
Harry first picks up his wand in Ollivander’s wand shop right near the beginning of the Philosopher’s Stone while accumulating supplies for Hogwarts. 
“A wand is the object through which a witch or wizard channels his or her magic. It is made from wood and has a magical substance at its core. Wands made by Ollivander have cores of phoenix feather, unicorn hair or dragon heartstring, and are of varying woods, lengths and flexibilities.”
As shown above, wands come in a variety of shapes, colours, sizes, flexibilities and materials. The main parameters that differentiate each wand are as follows: length, wood type, and lastly- core type. Each element is just as important as the last, and these numbered criteria are part of the reason why each witch/wizards wand is so UNIQUE. In fact, there are a considerable number of permutations. 
So, how is that that Voldemort and Harry ended up with the same wand core? What are the chances that these two wizards, whose fates would intertwine, ended up with phoenix feathers from the SAME phoenix.