Yes, you can use the same type for both ends of a driver. The transistors in the LED circuit are being used as emitter followers, which provides maximum current gain and no voltage gain and requires no biasing resistors. Current gain is exactly what's needed in that circuit, so using emitter followers for both ends gives simplicity.
But if you want to use e.g. all NPN transistors, you can use a common-emitter circuit for the pull-down side. In an emitter follower, the voltage follows the input voltage, while in a common-emitter circuit the drive is inverted - a pull-up current into the base will produce a pull-down current through the collector. The one extra thing you have to provide is a biasing resistor between the driving circuit and the base, to limit base-emitter current. It might appear to work without that, but it's a bad idea. You need to know the transistor's current gain so that you can provide the desired output current divided by that (plus some headroom) to the base.
This isn't just a digital thing - in analog electronics, emitter followers are used to provide current amplification for varying voltages, and common-emitter circuits are widely used to provide combined current and voltage gain.