No, it's sort of duplex (called half duplex). Duplex means that the channel is two directional at the same time. Like your telephone, you both can talk and hear at the same time. With a repeater, you have an input frequency and a retransmitted frequency. But when you key the mic, you can't hear anyone else on the repeater. So you are picking between one or the other half of the duplex channel. If you have a dual VFO radio, then you could tune both sides to the repeater output (say 145.145) and when you transmit on one side, the other would still listen. This causes feedback, though, if you have the volume up. But if you wore headphones, then you could hear your transmission and would be in full duplex with the repeater (not with everyone else using the repeater, though). The whole system, input freq., output freq. and all the gear comprises a duplex repeater. But each user station is only a half duplex since it must chose to either TX or RX and can't do both at the same time (with the exception of the two tuner radio example).
The reason you have two frequencies is that for the repeater to both TX and RX in real time it must transmit on a difference frequency than it's listening. If the repeater used the same frequency, then in real time it would always hear itself. Even with a shift you still have to isolate one side from the other to keep from causing problems. This is done with a duplexer with a single antenna.
It is possible to do a single frequency repeater (called technically a simplex repeater). This is usually done with a time delay. So the repeater listens until you are done talking, recording your transmission, and then rebroadcasts it after you are done. These are often also called parroting repeaters. These are used often for GMRS repeaters more than ham.
"Stand your ground. Don't fire unless fired upon, but if they mean to have a war, let it begin here." -- Capt. John Parker