Is this cheap "air conditioner" able to cool a room?
I doubt that it even has a cooling element, i suspect that it is just a fan + humidifier.
The fan+humidifier is the cooling element for this unit. It uses purely evaporative cooling to reduce the temperature of the system. It can do this because the phase change between liquid and vapour requires energy. By just passing a convective current of relatively dry air over a liquid water reservoir, heat is taken from the air to evaporate the water. This results in the humidified air being a lower temperature than before it entered the humidifier.
In this case, the heat doesn't just vanish. The heat lost is stored in the latent heat of vaporization of the water. If the vapour in the room were to begin condensation, the heat in the room would start to increase.
Basically, you're just using the humidity as a sort of thermal battery. You're able to store some of the heat in the room in the form of increased relative humidity, instead of having it go towards increasing temperature directly. The energy doesn't leave the system; it's just taken a different form as internal energy of the phase.
You can only remove so much heat this way, and the rate of heat removal decreases as the room's relative humidity approaches 100%. If you want to use that for constant cooling, you will need some way to remove the moist air and replace it with dry air (one that doesn't involve a dehumidifier that puts heat back into the room).
One this size is more of a personal cooler, placed right in front of you it will probably keep you a little cooler, it will do little to nothing to cool a normal sized room. But the principle is sound, as water evaporates it becomes cooler than the liquid water. Adding ice will cool the water, so the water vapor will be even cooler. Growing up in the 1960s, in Texas, all we had to cool our house were evaporative coolers (also called water coolers, or swamp coolers), These were large and blew a lot of air with a "squirrel cage" blower inside a box with vented padding on 3 sides which had water pumped over them. They were placed outside of a window so the humidified, cooler air was forced into the room. They would usually keep a large area comfortable even in the middle of summer(usually 20 to 30 degrees F, or more, cooler than the outside temperature). These work best in "dry heat" where humidity is low, as water can evaporate faster. They do not cool as well on rainy days or other times of high humidity.
It's an evaporative cooler: You fill it with water, it blows the room's air across the water, and the energy required for evaporation is heat that is thus removed from the room.
I don't know how well a small one like that will work, and in any case it is only going to work if the air is fairly dry; but in principle, at least, it is plausible and not a scam.