The word privilege, as it's usually used in these conversations, is rarely about the actions of the person observed to hold it but rather about how those around the person (or in this case couple) treat them.
If I were a man in my current job (military), I would have the privilege of knowing that when I walk into a meeting at least 90% of the other attendees will be of the same gender as me (I've been keeping a tally in my diary). When I need to speak to others at a similar level of authority I would have the privilege of knowing that I would be dealing with someone of my own gender. When I start a new job in a new office I would have the privilege of knowing that I will be judged by my actions rather than my appearance. If I were to become angry that my instructions to a subordinate weren't being followed and expressed my displeasure I would have the privilege of knowing that nobody would make jokes about menstrual cycles. When a new item of uniform clothing is issued I would have the privilege of knowing that it was designed to fit my body shape (seriously, I don't know what they measured when designing our old shorts, but it wasn't a human female).
None of that says that a male in my position is in any way lording it over me that he's male and I'm not or that he feels entitled to more respect than I get, but he's starting from a position of comfort that I'll never know. The fact that he has never asked to be treated any differently doesn't reduce the fact that, simply by being a male officer rather than a female officer, he is demonstrably treated differently in countless ways every day.
In the case of couple privilege, it exists whether the people in the couple use it to fuel their sense of entitlement or not. When an outsider assumes that the members of a couple will act a certain way and makes room for that to happen without friction then that's couple privilege in action. By asking people to examine their privilege nobody's saying that they should reject it out of hand or feel guilty about it, just that they should realise that not everyone has the same advantages.