Why electric pressure washers have a horrible noise like this: https://www.youtube.com/watch?v=AlxZYLeJeOA ?

To me it is rather normal sound of a piston pump.
To get the needed high pressure of a liquid at the used flow rates you are practically stuck with a positive displacement pomps. The dynamic pumps (centrifugal,...) have too high leakage losses, so become efficient only when pumping really high volumes (where the leakage is small percentage), but would be killing at flow rates for just a single cleaning nozzle.
Yes, there are other positive displacement pumps for liquid, but they require the pumped media to act as a lubricant for the pump mechanism. Materials that can be lubricated by water do exist, but are usually soft, so not suitable for a high pressure machine.
The linear piston is practically the only mechanism that could be lubricated by something else than the pumped medium (by the oil mist in the crank case), but the crank mechanism can not be 100% balanced, hence the noise...

It is just a shame that only piston pump is suitable for this. The noise of these pressure washers is rather horrible to me.
And Medved: Is a rotary vane pump or even a scroll pump, would be suitable for pressure washers?

Not for pumping water. The thing is the vanes or screw need to be lubricated, but the water does not lubricate metal on metal contacts. There are materials that are well lubricated by water (some plastics or woods), but those are too soft to be used for such a high pressure pump.

On the other hand vane pumps are common for pumping oil, they are used mainly because they offer the ability co control the displacement in a cheap and simple way, which is very useful in applications like hydraulic power steering or automatic transmission hydraulic control, or even in modern engines to provide the exactly required lubrication pressure without excessive losses.

Otherwise even simpler are gear pumps (traditionally used in engine lubrication systems or hydraulics powered from electric motors or so), but these rely on the pumped medium to provide the lubrication too.

I've seen schematics where gear pumps were used to pump lubricating water for propeller shafts on ships, but that is rather low pressure application (few bars; just to overcome the hydrostatic pressure of the water outside of the ship so no dirt could flow into the bearing from the outside; the water is filtered before using as the lubricant there), so using plastics for the pump gears make the water good enough lubricant there (same as the prop shaft bearing itself). But those plastics are by far not strong enough for the higher 10's bar pressures needed for the washers.

What about scroll pumps? And btw: Water pumps in pools and buildings with independent water infrastructure like in the US, I don't think these uses reciprocating pump like pressure washers.
Also: Isn't in pressure washers, the pump increasing the water pressure and not simply pumping it?

A Scroll Pump is used to compress gases.

Pumping any liquid means increasing its pressure (or better to say creating a pressure difference; but because the inlet pressure can not be negative, with pressures of a bar or above the difference practically means almost the same as the output pressure), otherwise there won't be any "motivation" for the liquid to flow (against its viscosity friction).
From one side you may say by forcing the water to move means you have to overcome a pressure, from the other side you may side creating a pressure is what made the water flow.

The conservation of energy in pumping liquid means the power gets into a form of creating certain pressure difference times flow. So with a certain power you can have either high flow rate (= volume pumped per unit of time) at lower pressure, or a low flow rate at high pressure difference.
What makes the pumps different to each other (even at the same power level) is what pressure difference they must pump vs what flow rate they pump. For the home running water you need about 2bar pressure, if it means pumping from 30m deep well it becomes 5 bar (for the pump inside of the well) but flow rates about 1..2 l/sec, for pools (I mean the various water slides or so) the flow rates are 10 l/sec, but pressures about 0.5 bar, but for a pressure washer you need 50-ish bar, but with flow rates in the 0.1 l/s ballpark.
For the 2 bar a centrifuhal pump (no contact between moving parts, so no lubrication needed) is fine, 5 bar in a narrow thin format use to be scroll pumps (steel rotor in a rubber/plastic stator, lubricated by the pumped water), but the 50 bar for the pressure washer is just out of reach for anything else than a metal. And metals means it needs oil to lubricate, so it has to be separated from the water. And because that separation could be reasonably made only for a reciprocating mechanisms, with limited or even none sliding surfaces between the oil vs water (often diaphragm separation is used between the oil and water, the oil below the membrane is pumped up/down by a physical metal piston, the membrane then forms a kind of separating floating piston between the oil and water, so there is no contact between the water and oil and no sliding surface is left without any proper oil lubrication).

A Scroll Pump is used to compress gases.

It depends on the geometry of the scroll.
For gas the scroll should be made so the pockets contract as they are "pushed" along the path, following the way how the gas is being compressed (like the radial one you have shown).
For pumping a liquid the pocket volume stays constant (as the liquid is not compressible; often used for the thin rod format pumps for water wells, the rotor is a "snake twisted" steel screw rod in a rubber sleeve, the pockets are then formed between the snaking turns and the sleeve, pushed along the axis as the snake rotates; for the flows and pressures used for the well water it results into a narrow long format, so fits easily into the thin drilled water wells).

Either way there are surfaces sliding against each other in some form, so need to be lubricated. With gas compressors usually an oil is mixed into the gas on the inlet (either injected or forming mist with the gas), then the resulting mist is entering the main compressor, where the oil settles on the surfaces and lubricates them. On the discharge the oil is either separated to be cycled back (air compressors,...) or is just being passed with the medium through the system and returns with it back into the compressor (most often in refrigeration systems, where the oil is then also lubricating the various control and regulation valves along the refrigeration system).

When scroll pumps are used with liquids, they have to rely on the liquid to provide the lubrication. For the water pumps the used plastic/rubber is intentionally formulated so the water is lubricating it against the steel scroll, but because it is soft, it can not handle more than about 10 bar.

Update: I've seen today a pressure washer that its pump have a significantly different noise than most household ones with piston pumps. is this noise is also of a piston pump as well?
https://www.youtube.com/watch?v=j8NnLRqOZXg

It is actually possible to make a piston pump quiet, it just gets a bit more complicated (multiple cylinders, complex balancing, advanced silencing,...), therefore more expensive and often also less efficient (for the same pressure and flow require larger power).

Also someone may have used some rotary pump format, if we are talking about some lower pressure models (in the 10 bar ballpark).

In this video, the pump noise frequency isn't 50hz. Also, The turning on and off of the pump isn't instant like most pressure washers, but gradually.

That could just mean either something like an universal motor, or something electronic (either induction on an VFD, or even some permanent magnet BLDC - to compensate for a lower pump efficiency by using a more efficient motor).

Also this can mean they just put the pump somewhere inside their booth, so the noise is not heard outside