The account has his name on it. It is his account.
Sure. And it’s his face on the billboard, and so on. When your name and face are regularly handed off to others to create a message to associate with it, sometimes the best you can do is pull the offending material after-the-fact and presumably identify/deal with the person who created it. If you consider Twitter passwords to be sacrosanct things only to be given to people with the highest vetted clearance, you would be (perhaps properly) horrified by how it works every day in much of the world, where the intern gets handed it on day 1.
If it is the case that any new intern gets handed the password on day 1, I would suggest changing the name of the twitter account. Maybe change it to “Ted Cruz campaign” or “The Office of Ted Cruz” or “Ted Cruz INC” or “The Staff of Ted Cruz”. Something that indicate it is a twitter account belonging to his group(what it is called), and not his own private twitter account.