David Vonderhaar has confirmed on Twitter that there will be character customization in Black Ops 3 mutliplayer, and he has teased a look at the different parts of the character we can customize.
@RobsStang You should perpetuate the myth that their is no character customization in MP and just ignore everything I've said about it. +
— David Vonderhaar (@DavidVonderhaar) October 6, 2015
To confirm, there is character customization in both campagin and multiplayer in Black Ops 3.