Why would boiling milk in an electric kettle break the kettle?
The electric kettle is (clearly) not designed for this. The main issue is that milk doesn't evaporate, whereas water (obviously) does. The secondary reason is that milk will burn.

Milk is a complex mixture of water, fats, and proteins. The fats and proteins will separate out from the water when heated, and form a layer on top. Unfortunately, this layer prevents the water from evaporating - it traps it. This is what causes milk to boil over. Incidentally, the reason potato or pasta water boils over is due to the starch.

The way kettles turn off is by steam reaching the top of the kettle, rushing down a tube and causing a bi-metallic plate to expand unevenly, tripping the switch. 

No steam means the kettle will never turn off. Because the kettle doesn't turn off, the element continues to heat the milk next to it. With water, the hot water will rise, rather than stick to the element. This means leaving the lid open won't cause a fire - at least not until all your water has boiled away. Unfortunately, milk will burn, and this layer of burnt milk will prevent effective heat dissipation - the milk in contact with the element is not moving throughout the remainder of the liquid. This causes the element to get hotter than it is meant to.

However, kettle manufacturers have thought this through - they don't want their products to catch on fire (well, most don't) - even when they are misused like this. They will have built in a small (one time) temperature switch, like this one. This acts a little like a fuse, but for heat, not current. They are often called "thermal fuses" for this reason. When the element reaches a temperature which has been deemed "too hot" (probably around 190ºC, perhaps a little hotter) this switch is tripped, and current can no longer reach the element. I attempted to get a measure of the temperature of the heating element of a kettle, but I could not get good enough contact between the element and my probe.