First and foremost please please please remember water is F***ING heavy.
I know it sounds stupid to say but it's so easy to forget how fast it adds up.
Is your shed wood floored?
What is it sitting on?
Would you be comfortable grabbing a 50cm diameter boulder and placing (Not dropping) it on the floor and not have it buckle let alone break?
If yes then you might be alright.
MIGHT!
is the outlet 110? Check the fuse and not how many amps it's rated for.
If it's just you and you have at least 20 amps I would seriously consider one of those inflatable hot tubs.
You can get one from anywhere between $200 and $600 and they will work for that wattage AND distribute the water weight enough you won't crack a concrete base, even a not too thin one.
If you are thinking of a cast iron tub or something similar maybe with higher walls or smaller footprint as the shed isn't big enough for a hot tub then you need to seriously worry about that it will be sitting on.
Finally dumping the water once in a blue moon with 2 squirts of soap isn't great but it will do minimal damage but 2 or 3 times per week is a LOT of water that will concentrate in a small area and if you plan on washing at all that adds to a lot more soap and detergents then you realize concentrating in your lawn.
Consider doing some filtering before you dump. Charcoal or sand to start and that will help quite a bit. It'll mean very slow drain but you don't really are about that do you?
DIY hot / bath tub in shed?
This is probably a dumb one, but here goes....