We still don't really know how Equestria's run really. Like, we don't ever see the princesses really making any decisions except in emergency circumstances — they either pace around worrying or blow shit up. Is there any elected official(s) that they have to answer to for day-to-day or that handle the standard "keep the country moving" thing? Who knows. Who the hell even were the officials they met? Can't have been that special a visit if it was over that quickly.
Either way, being a royal in Equestria doesn't seem to have too much difference to being royal in Britain (no real impact on how the country's run) — minus the whole god-like magic powers thing. Other than that, basically the same.