I'm not sure.

He's certainly the Commander of the Avengers groups.

He does not seem to have a support structure paramilitary force, but remember, its "early days" right now, and nature abhors a vacuum. I find it hard to believe that any President would just mothball all the infrastructure and give pink slips to trained operatives; talk about a recipe for disaster!

Then there is the old "Champion"-level clearance, which IIRC superceeds and is superior to the normal chain of command; kind of a rank of various kinds.

Rogers' rank of Captain seems to be just as spurious as his "cover" rank of Private.