Why can the Sony A7R II overheat, and can firmware really fix it?

All information below is general in nature and may or may not be specifically applicable the the Sony α7R II.

Why does α7R II overheat?

Any electronic device overheats because it generates heat faster than it can dissipate it in the environment in which it is operating and eventually the internal temperature reaches a point that the hardware no longer functions correctly.

Is it solved?

Not according to the report you cite in the question.

Keep in mind that many more cameras than just the α7R II will tend to overheat when used for a while in direct midday sunlight as described in that post. Even when the ambient temperature is not terribly high, a black camera left exposed to direct sunlight for very long will be heated to considerably higher than the ambient temperature. I've had it happen with Canon bodies on occasion where the ambient temperature was fairly high and the camera was left sitting under direct midday sun.

How is an electronic heating problem supposed to be solved by a quick software update?

In general, by either:

• Limiting the maximum performance of the hardware components so they do less work per unit of time and do not generate as much heat in the same time period, thus giving the camera more time to dissipate the heat

or

• Updating the actual code to require fewer processing steps or less intensive processing to get a similar result

or

• A combination of the two above methods.

How successful such a method might or might not be depends on how much the amount of generated heat per unit time is reduced, combined with things such as ambient temperature that affect how much heat may be dissipated over that same amount of time.