So, I'm playing the game, many hours in and I go to House and give him the chip, he upgrades his robots and tells me to kill Caesar. So I go to Caesar land and talk with the dude, I have decided that I'm gonna do his quests this time around. So Caesar tells me to go to the bunker and destroy House's robots and stuff. However Caesar gives me the chip to get into the bunker to get to the stuff I'm supposed to destroy..... hmmmm something isn't right here, isn't there only one platinum chip? How did Caesar get it from House and give it to me? This kinda sounds like a "continuity error" if you ask me. Kinda funny though.

Well, remember, the Legion does take all your weapons away from you once you land on their shores.

I can't remember, but do you get the chip back from House after he upgrades the bots? The Legion easily could have taken it from you over at the Fort along with your weapons. And then Caeser would have given it back to you.

Caesar charismatic skills loot your platinum chip and gives that same platinum chip back using his ultra-psychic abilities...A mole rat told me this...

If you ask the guy who takes your weapons and "banned items" at the entrance to the Fort what is considered a "banned item" he'll tell you that anyone who isn't legion doesn't get guns, but no one is allowed chems, but also sais that this rule "extends to the platinum chip you're carrying for now".

If you actually go into the fort, prior to obtaining the chip, you can get it by killing Caesar.
I don't know if it is pickpocket-able though, I didn't even realize Caesar had it before I began my murderous unarmed rampage.
You can then obtain a second chip from Benny.