A false awakening is still a dream, yes. It is a very vivid dream which begins with you experiencing 'waking up' (but you're actually still dreaming). Everything feels very realistic and quite often the dreamer truly believes they have woken up (I had one this morning that caught me out). The dreamer may spot something a little 'off', for example a piece of furniture out of place or a strange coloured sky that prompts lucidity. But generally these types of dreams are very convincing as the first thing you expect to do after a dream is wake up, and without a certain level of awareness false awakenings simulate just that.
Dreaming is my drug