Why does boiling water in the microwave make a cup of tea go weird?

I doubt that it's superheating, as I understand superheating is a rather violent phenomenon. The most likely explanation is dissolved gases; gas solubility in water decreases with temperature. The kettle boiling process is very turbulent and so can release all the dissolved gases, resulting in pure hot water. The microwaved water is heated in a very gentle way, and so the gases do not yet have a chance to escape from the surface... not until they can start nucleating on your tea bag.

A common problem with microwave is that you lack control of the temperature of the water. Second problem is that the water is heated from the top and the sides of the mug mostly. Result is, that the content of the mug is not really boiling hot everywhere, altogether (after mixing) it is well below 100 °C A side effect from this is, that the water is not dearated like it is when You boil it on a hot plate. So: You have some water close to 100 C at the surface and colder (containing air) water below. When You dip in the bag, all that mixes, the air bubbles out.