Actually this is correct, as far as it goes. Military members are not supposed to do politics while in uniform. They also cannot run for office while on active duty. (some exceptions for activated reserve/guard). They can ID themselves as military members, but must make it clear they speak only for themselves.
The rules apply mainly to *partisan* political activities though, and so their application to issues oriented events like this is at least questionable.
Still the moderators reaction was also out of bounds, way way out of bounds. It's none of his business, the military member doesn't answer to him, but rather to his branch of service, and of course to his commander(s).
Wasn’t some vet just prosecuted earlier this year for wearing his uniform to an anti-war rally in DC? He was going to be courtmartialed and dishonorably discharged, and he didn’t even speak. He was just wearing his uniform to a political event.
Apparently it’s against Army regs to wear the uniform to any partisan or political event, whether it is affiliated with a particular party or no.