The salesman said....
You know why they (Salesmen) wear name tags (per the old joke) so they can remember their name.
The fridge TECHNICALLY runs off the inverter plugged in or not...
But what is the INVERTER running off of when you are plugged in.
Inverters come in two "Flavors" for this post (really 3 but we will call the 3rd 2A)
Stand alone
Battery---Inverter---Device(s)
And pass through
LIne--Inverter--Device(s)
-Battery+^
and Inverter chargers (Same diagram as Pass through but the ^ becomes ^V works both ways)
You likely have the last of those.