Consider the humidity in the shed and ventilation.

Did you know you can buy inflatable hot tubs?

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?

Does the shed have a concrete foundation?

Sounds very doable! My friend has an old claw foot tub that he lights a fire under. If you want something a little less country, you can buy on demand electric or propane water heaters and hook your hose up, though I'd expect the electric one wouldn't be able to keep up at 120v. Hardest part of this project is probably moving the tub. I say go for it!

The concept is doable for sure, but I'd reconsider the draining issue. The volume of a hot tub is quite something, and if you're using any kind of soap or other chemicals, you've got a potential environmental challenge on hand. Even if you only use biodegradable stuff, that doesn't mean that plants will rejoice being smothered with soapy water.

For the heater you'll need something with a decent wattage to get it up to temp in short enough time, so make sure you're not overloading the circuit. Those outlets in some shed are often not the best.

Propane/butane would work as well, but in that case you'll need to take care of proper ventilation as well. Or just leave the door open.

I’ve seen either a video of someone having done exactly what you’re talking about (including the prefab shed) or maybe it was a pic. So yes. Go for it!