Battery wear on small solar powered devices like watches or pocket calculator
My "analog" Seiko Solar watch uses a rechargeable battery  V157-0AA0 according to SEIKO website.



Since there is no self-heating or high current, the wear factor is extremely low.

You can see the conditions required for operation.  Sor so many years, I have had a failure since receiving this birthday gift.  The same with my SHarp solar calculator, many decades old.  It always works forever from ambient interior light.  I have less kind words to say about my expensive mechanical wrist-action watch.


  Do they have a chip programmed to either use solar or battery, and only charge the battery if it's rather low?


No.

It runs off the battery and the Solar charger manages to maintain sufficient and not excessive charge.


  when the energy stored in the rechargeable
  battery is reduced to an extremely low level, the second hand starts moving
  at 2-second intervals instead of the normal 1-second intervals. (Some calibres have no such function.) The watch remains accurate even while the second hand is moving at 2-second intervals.
  
  NOTE ON POWER SUPPLY
  The battery used in this watch is a rechargeable battery, which is different from ordinary silver oxide batteries. Unlike other disposable batteries such as dry-cell batteries or button cells, this rechargeable battery can be used over and over again by repeating the cycles of discharging and recharging.
  
  The capacity or recharging efficiency of the rechargeable battery may
  gradually deteriorate for various reasons such as long-term use or usage
  conditions. Worn or contaminated mechanical parts or degraded oils may
  also shorten recharging cycles. If the efficiency of the rechargeable battery
  decreases, it is necessary to have the watch repaired.