Old fridges have a compressor which uses a fair bit of power on start up, it needs to start before the pressure builds up, and if switched on too quickly after being switched off then an over load will trip and give it some time before resetting that allows the pressure to drop, if this over load is used too much it will fail, so volt drop with any refrigeration unit using a single phase motor is important, and there is some volt drop across any fuse or extension lead, so manufacturers recommend you do use them.
However in most cases the volt drop is not enough on a short extension lead or multi adaptor to matter, so likely not problem, but if you hear the fridge having to re-try starting then there is a problem, but not found an fridge with the problem AC units yes, fridge/freezer no. But there is always a chance.
Modern refrigeration units don't use single phase motors, but use a multi-phase with an inverter, and the inverter corrects for volt drop and the start current is much less, so they have no problem with volt drop, but it still has same warning in the instructions never seems to have been removed.