So fuel goes off after a month or so and can damage your engine. If they never run the fuel, do they add stabilizers? Do they just accept the reduced engine life? Or do they run with an empty tank?

AIUI most plugin hybrids have sealed pressurised fuel tanks, which prolong the life; many will also automatically use the engine such that they consume a full tank once every six months or something even if battery power is always available.

I use ethanol-free (i.e. 100%) gasoline in my hybrid; the gas tank is also pressurized. These combine to give shelf-stable fuel, for years.

Where do you find ethanol-free gasoline? I haven't seen that in decades.

We had it available here until a few years ago... I think you have to buy up-grade gasoline now to get it without ethanol.

Not sure where "here" is, but counter-intuitively (in US and much of Canada - where it has to be labeled) mid-grade is just low-grade + ethanol (which boosts octane), so it's often a down-grade over low-grade (important when you run a carburetor). Grade.

Canada, but I should have checked the current state of affairs before posting - it actually looks like no grade is no regularly available without added ethanol.