Does my electric stove heat up quicker on the highest gear?

Yes, it will bring you to a boil the quickest.

Many heating systems we encounter in everyday life have thermoregulation, and it is counterproductive to run them at full blast at the beginning. For example, an AC unit will "know" on its own to run all the time until it reaches the target temperature, and then start turning on and off as needed to sustain it.

Stoves are different. They don't have any temperature feedback, and all you regulate for them is the total output of energy fed into the hob. It doesn't matter if their lower levels are solved by going off part of the time, or heating constantly but with less energy. They are not even supposed to reach a constant temperature, and for any given stove mark, the final temperature in your pan will be quite different depending on factors such as the pan material and the amount of food you are heating. Some fancy new stoves try incorporating a sensor below the plate, but the one I have used was so far off that I found it useless and cooked in "constant energy" mode with it.

So yes, whatever the technology of your stove, your water will boil quickest if you use it on the highest setting.

It may depend of the manufacturer. On every electric stove I have used, the heating element operates similar to a microwave oven. The control technique is called ‘pulse width control’. This means the element is either on or off, and the percent time that it is on is set by the control knob. For example, a 10 could mean the element is on 100% of the time, and a 5 could mean that it is one for 30 seconds and off for 30 seconds. In your case, I would assume that the 6 setting would keep the element on the most (maybe not 100% of the time), and by turning it down to 5 would increase the amount of time that the element is off.

The more the element is on, the more heat it is putting into the pot, and the faster the water will boil.