Would running a generator on its side DAMAGE it? Talking about any Honda generator?

Seriously, how much damage could it possibly do?

How much damage? Now lets see. Melted bearings. Check
damaged crankshaft - check,
Scored piston- check
broken piston rings - check
ruined cylinder - check.
Chewed out camshaft - check
metal fragments throughout - check

Oh wait. The bolts that hold the crankcase together won't be damaged and could be reused on a different engine once you throw this one away.

PS I answered another question about a 400 w fridge.
You suggested a 350 W generator would run it within the comment.
I must warn you that the generator must not only supply the 400 - 500 W that the motor uses when running it must ALSO supply enough to get the motor started.
So I suggested that a minimum of a 1000W generator would be needed.
Unless the rating on the fridges compliance plate is not listing the running figure.
My fridge is a lot smaller. It takes about 100 W during running but almost 300 W during the moments that it is first turning on.

Inadequate power available means that the motor "Stalls" and can burn out if there's not a thermal overload cutout fitted. IMO yes, as the oil in the sump wouldn't be circulating as designed.

"how much damage could it possibly do?"

One heck of a LOT, meaning permanent, irreversible damage. The oil pump pickup is designed to work with the engine sitting on it's base. Turn it on it's side and the engine will soon experience oil starvation and destroy the bearings rendering the engine useless. Same goes for any engine other than an aircraft engine which is designed to be flown on it's side and even upside down. The cost to do that is too great for a car, lawn mower or generator engine. The oil needs to stay in the bottom of the sump where the oil pump can get to it and lubricate the motor. Not on its side.

IMO yes, as the oil in the sump wouldn't be circulating as designed.

"how much damage could it possibly do?"

One heck of a LOT, meaning permanent, irreversible damage.

The oil pump pickup is designed to work with the engine sitting on it's base. Turn it on it's side and the engine will soon experience oil starvation and destroy the bearings rendering the engine useless. Same goes for any engine other than an aircraft engine which is designed to be flown on it's side and even upside down. The cost to do that is too great for a car, lawn mower or generator engine.

The oil needs to stay in the bottom of the sump where the oil pump can get to it and lubricate the motor. Not on its side.