We have a bumper crop of people out there with a James Bond complex.
I see this everywhere. If a guy was an electrician on a SOF helicopter, he'd take on an 'Afgahinsatan Operator' persona and act like he's Delta.
I have a guy at work who worked in space operations staring at a scope, but he has a full beard, long hair, and Spec Ops ball cap...in an office building.
Army and Air Force seem a bit different, from the outside looking in, wearing the hat of a unit you were a part of seems fine. I believe the Army has tabs and pins that would further distinguish operators from support personnel. If he is only wearing the command cap, and he served in that command, I am not sure that is an issue. It is an issue to vets, if you NEVER served in any capacity in SpecOps and you wear a hat with a SOC logo. A.k.a A-Waltz.
Assigned as support personnel to Recon and a SOC was great, awesome commands, excellent senior enlisted and officers, I am proud of my time with them. Maybe that is what is going on with your coworker. Then again there are those who like to give civilians a false impression of their service.
It’s everywhere... so idiotic.
A far cry from the WWII, Korea and Vietnam guys who came home, and dressed clean cut and went to work... and never said a word about it.