How exactly do you check if something is authentic? I can have a hundred Theranos-branded t-shirts, mugs, pens, diaries, socks and whatever else you want made and shipped to me in like 2 weeks.
It would be trivial to prove you worked at a company if that's how you were given it, you might even have photos of an event where stuff was given out, emails detailing a "welcome pack", I'm guessing it's these sort of things that are used as proof.