Pokemon such as Mankey and Abra gain items after evolution. Primeape has metal rings around its wrists and ankles, and Kadabra gets a spoon. What I'm confused about is that Braixen gains an item during evolution as well, the stick on its tail, but in the anime it is shown that Serena's Braixen broke her stick and found a replacement afterwards. So are the items gained during evolution part of the Pokemon? If not, does it count as a held item?
The items were most likely not intended by the developers to be a part of the Pokémon, but rather, a part of the Pokémon's design. For instance, the entry on Braixen from the Pokémon X game is:
It has a twig stuck in its tail. With friction from its tail fur, it sets the twig on fire and launches into battle.
So, do all Braixen have twigs? Yes, they do, but this is just because it is a game, and that is how Braixen was designed. And the anime reflects the designs in the games, and sometimes the anime tries to give some reasoning as to why the Pokémon have these items. But is this twig a part of Braixen? No. As evident by the Pokédex entry, the twig is just something that is stuck in its tail for whatever reason. It is also evident in the anime because they were able to replace it when it was broken in the episode Mending a Broken Spirit! (Japanese episode title: A Broken Twig: A Broken Heart! Braixen's Strong Feelings!!)
Edit: If we reference Pokédex entries, we see that they tell a narrative that's a little different from what we see in the games and in the anime. For instance, from Braixen's entry, I think the narrative is trying to say the twig in it's tail is just a twig it picked up on a stroll through a forest. If we look at what actually happens in the game and anime though, the twig magically appear as soon as Fennekin evolves. The point I'm trying to make is that there is a lot of hand-waving going on when a Pokémon evolves.
If you'll also recall in the first season of the Pokémon anime, Ash's Caterpie did not evolve in the same manner as all subsequent Pokémon. To become Metapod, it encased itself in a cocoon using string shot, and to become Butterfree, it broke the shell. This means of evolution makes more sense, but it wouldn't be feasible to come up with a unique means of evolution for every Pokémon. I suspect that the white glow method was adopted from the games as a convenience since so many Pokémon were going to evolve during the series and it introduced this inconsistency with items appearing out of nowhere.