I feel like an
could also be a right answer.
I cannot walk, but I can swim.
Eye in the fluid in your eye socket.
A liquid keeps me from being slim.
Eye is kept big and round from liquid in the eye via osmosis
Only few people can see my wave.
REM sleep? The eye does literally go up and down so is it waving?
People can see if they open your eye while you are asleep or by using
some kind of a scanner. Also REM is often literally plotted as a
"wave" (sine wave, etc) on a graph.
I spend my nights in a kind of cave.
You close your eye lids. Need I say more?
I don’t breathe, yet cannot drown.
Eyes don't breathe, and are in liquid.
When things get serious, I prefer upside-down.
The eye actually takes in light upside down and your brain corrects
it. So if you take serious as being awake and seeing, then everytime
your eye is open it is actually taking in everything upside down.
Sounds pretty serious to me.
I put forth the eye!