But our consciousness is always on while alive.

Not all the time. Sleeping is the obvious exception. You may quibble about whether REM sleep counts as “consciousness”, but there are a couple of deeper types of sleep you cycle through that go way down into inactivity.

There’s also total anaesthesia, which (depending on the particular type) can shut your brain right down deeper than sleep does.

Then there’s people who have clinically died and then recovered, including some record-holders with Lazarus syndrome and who drowned in cold water - the record there is a 2-year-old who was submerged for 66 minutes and had a core body temperature of 19 degrees C when she was pulled out.

Within Star Trek itself there’s also Cryogenics (Khan and company were frozen while traveling in the Botany Bay) and Cryonics (the frozen people who were revived in TNG’s “The Neutral Zone”). Were those people still the same people as they were when they were frozen?