To my knowledge it's physiologically not an altered state of REM as it is just regular REM sleep. Hypothetically, you can have a lucid dream just as you would a regular dream, and correct me if I'm wrong, there's no real difference from your body's point of view. Maybe parts of your brain is more active, since you have conscious awareness.

I can't speak for the parent poster, but maybe they meant that you tend to have more lucid dreams (and also remember your dreams altogether) whenever your sleep is broken up more during the night. In other words, lucid dreams don't necessarily interrupt sleep, but interrupted sleep tends to make them more likely (at least in my experience).