It's actually explained in a scene in The Phantom Pain.
Eli: "So what's with the robot arm? Even that Miller guy chooses not to look so silly. 'Come on Boss, don't you have a spare, I lost an arm too'"
*Ocelot starts laughing*
Ocelot: "Where did you get that arm from anyway?"
Big Boss: "This relic? We have our old Soviet friend Zadornov to thank for that"
Ocelot: "Well, you sure as hell won't ever catch me with one. You need a real arm to use a revolver. What say I take yours, Eli, if I ever lose mine?"
Eli: "Hey!"
*Ocelot and Big Boss laugh*