A refrigeration plant be it fridge or freezer has a motor, and unless inverter controlled it will have an inrush, so although the unit is rated under 100 watt, it can draw more like 1000 watt on start, and the speed at which it starts is important, it needs to reach speed before the pressure reaches a point when it would stall the motor, just in case the units have an overload, which auto resets, so if for example it is unplugged then plugged in again before the pressure has dropped, the overload will trip, and it will take some time before it resets, by which time the pressure should have dropped.
The overload is not designed for repeated use, so if it trips many times, they tend to burn out, and the volt drop caused by using an extension lead, both with the lead and duplicating of fuses, could be enough to cause the unit not to start first time.
So most fridges and freezers simply say do not use an extension lead. In real terms you would be very unlucky for the volt drop to cause repeated tries to start, and one would hope if it did happen you would notice, but the manufacturers put the same warning on inverter drive units, with an inverter drive you have a three phase motor, these start much easier, and the inverter limits the current on start so you don't see much of an inrush, with mine using an energy meter I can see when the defrost heaters cut in, which are around 140 watt, where motor more like 60 watt, so with an inverter drive there is no real reason not to use an extension lead. But the manufacturers have not changed their advice.
In the main only problem is after a power cut, where loads of devices are trying to start together, and the overload would take care of that, so you would be rather unlucky to have a problem, but forewarned you should hear when the fridge or freezer has to try multi times to start.
In real terms the only time I have seen a problem was in Algeria, an electrician, well he said he was an electrician, thought if he used two 110 volt phases he would get 220 volt, since as name suggests they are not in phase, the actual voltage was 190 volt between phases, and the air conditioning units kept failing, but there was a 30 volt under voltage, so hardly surprising.
So I agree with
@JohnW2 but have given you the back ground why they say don't use an extension lead.
