It is not a republic either. At least it is not a republic in a way that the Greeks, Romans, or American Founders would have understood it. It was meant to be a republic, but it is no longer.
The US is actually one-half mandarinate (rule by a permanent government comprising the civil service and academia) and one-half spoils-system kleptocracy (rule by organized factions who get the government to give them money or favorable law). Elections are used mainly as a legitimacy mechanism. You can think of them as almost a religious ritual, like taking communion. Elections are a way of giving people an illusion of control, and of proving overall buy-in in the system, while in reality popular influence is kept quite limited.