Well I don't think Rhodes is on the same level as Captain America. Cap was originally a mascot for propaganda. While word of his exploits once he did get into the field presumably made it home (there was a medal ceremony so he was definitely acknowledged, & his victories would have been very good for morale), he'd probably have been seen as something separate (I mean, he's seen commanding other soldiers, so did he get officially promoted to Captain during the Fight Allllll the Hydra montage? Or did they just accept it when he ordered them about they just sort of assumed he had the right to command them?), & he probably acquired a sort of semi-mythological status after apparently dying & few civilians having seen him IRL (aside from the propaganda shows).

Rhodey meanwhile is a soldier first & foremost. He has an actual official rank that he earned the normal way, an actual position within a military department, & he's seen doing spokesperson work at least once so people outside the army know his face & are vaguely aware of his existence as an actual person (also propaganda work, but a more modern, subtle kind). He just snagged himself his own Iron Man suit, which technically would qualify him as a superhero, but the difference between him & Tony is Tony has made it clear that he is independent & is trying to divorce himself from his weapons-dealing past, while Rhodey is a soldier who has used his acquisition of a suit to continue his role. So while people evidently see him as a sort of superhero (you get the impression that Harley seems to, at least) he is still firmly tied in to the military in their minds, so they sort of just go along with it.