Many systems, especially older ones, are set up this way by default.

But you can get, for extra $$, a switch to disconnect you from the grid entirely when it's down and run from your own solar power / battery. People who live in cities with underground wires normally don't bother, but it's essential in the countryside (IMHO).

Note however that many people have only maybe 5kW or 8kW or something like that being added to grid power by their solar setup, so if there is no mains power then it doesn't take many 2kW appliances (microwave, kettle, clothes washer (when heating water), dish washer (ditto), hairdryer, vacuum cleaner) to overload it. Not to mention 3kW hot water heater or 3kW+ stove oven.

I have a 3600W off-grid system (Pecron E3600LFP) and I run pretty much all that stuff from it. I added up and I could try to turn on 14kW of stuff at the same time. But I don't, obviously.