Does Inverter refrigerator need automatic voltage regulator?
An uninterruptible power supply or UPS will protect the electronics in your fridge, however a fridge has a fairly high peak current at times so the UPS needs to be quite large (one for a computer won’t have the wattage to support the fridge).
I would recommend getting a whole house surge protector and installing it at your service panel (breaker box). With a whole house surge protector everything in the home is protected from spikes. Brown outs are not as bad with most inverters as they are smart enough in most cases to be able to operate down to 90v with no problems, if they can not function at lower voltages they just shut down for low voltage on their internal (usually) dc buss. 
So do you need a UPS, probably not, but I would suggest a whole house surge suppressor especially if you live in an area that regularly has lightning and or heavy industrial loads close to your location.