Everything remains from Kyurem: IV's, EV's, Nature, Shiny or not, Nickname, Pokeball caught with, the only thing it get's from the other dragon is the signature move, I don't remember which one: Fusion/Blue Flare, Fusion/Bolt Strike.
Kyurem always inherits the EVs, IVs, nicknames, shininess, Pokeball, OT, ID, etc. of its own. All that changes will be it inheriting Reshiram/Zekrom's signature move Fusion Flare / Fusion Bolt.
Therefore, your Kyurem will retain its own OT and ID, and not of Zekrom's. I also tried experimenting this in-game by fusing two different ID and OT's Kyurem and Reshiram, as well as asking this question in Bulbapedia's discord server. They replied that it will inherits it's own, and that happen in-game as well.
Hope this helps!