Does the Play with Pikachu/Eevee function do anything?

I've found that showing your partner Pokemon attention will give them special abilities in battle. According to serebii.net, this could include: "[increased] Critical Hit chance, ability to remove status conditions and hold on in battle." I believe "hold on" here means the ability for the Pokemon to survive a strong attack with 1 HP left. [1]

Additionally, after showing your Pokemon attention for some time, you will receive gifts (items) from them:

From GamesRadar: "As you bond with your buddy, it'll unlock various perks, the two most important being a pair of sure-hit moves that you can use in battle. If you're fighting with Pikachu, a quick shake of the Joy-Con or Poke Ball Plus when prompted will cause it to unleash a sure-hit attack called Pika Pow. Doing the same while fighting with any other Pokemon will cause Pikachu to use the simply named Pikachu's support, which improves the all the stats of your attacking Pokemon."

"Playing" with your buddy will allow you to give them new hair styles!

If you ruffle up their hair with either 1 or 3 fingers (in tablet mode at least) you can cycle between different haircuts depending on how many fingers you used.

It is also possible to give them an afro by using all 10 fingers. Its difficult to do with adult sized hands but the easiest way I found was to hold down 1 finger first to enter the "ruffling" stage and then try to fit all 10 fingers on the screen and move in the same direction.