I rent my apartment, and it comes with a fridge, so I guess I'm renting the fridge too. IS this not common? Do people usually bring their own fridge in when they rent?

I think you mean "Few people elect to do the thing they barely know exists."

The model being peddled on TV/radio/etc isn't "Rent a fridge." is it?

This is just another "language war". The airwaves and communications channels are monopolized by discourse that protects the known way of doing business. It's not a law of physics it be that way.

Because it's more expensive over a relatively short period, perhaps?

I suspect for long term goods that are below some cost, it's just cheaper for people to own them vs renting over the lifetime of the device. A corporation will tend to finance the capital cost of the good if it's renting, then needs to pass on the interest costs of those goods to the renters. On top of that a rental corp also need to maintain techs to install/remove appliances. It all adds up to higher cost to rent than own in this area.

A company will also pay for and pass on the cost of non-paying customers in terms of repossession or loss of rented units.