Where did the anime trope of "clothing damage"/"clothes being torn" originate? I noticed that commonly now in anime that mostly female characters' clothes are being torn especially in fight scenes.

I googled and found this to be called Clothing Damage, and here are a list of examples from anime and manga from which I remember in High School DxD where the main protagonist Issei uses a technique called Dress Break to make his female enemies lose the will to fight. Also found in Dog Days and Freezing.

Even JRPG games such as Doki Doki Majo Shinpan! and its sequel, Doki Doki Majo Shinpan! 2 Duo has this.

Who started this trope? Is it somewhat related to the origin of the tentacles trope in anime?
The most likely culprit is Go Nagai, the father of Ecchi fan service. You can probably trace it down to as early as "Shameless School", which often had female characters "losing" their clothes:



But the actual tearing off of clothes may have first been in Mazinger Z's manga (circa 1972), linked from the very article in the question:


  
  It happens to Sayaka Yumi in the original manga sometimes (too many times for her liking). Given that it was created by Go Nagai, the man who introduced Fanservice in anime, it is not surprising.
  In one chapter, several female robots with blades replacing her arms sliced her dress to ribbons, without actually cutting her flesh or underwear.
  Great Mazinger: Jun Hono suffered from this more often than Sayaka (much to her disliking).
  It also happens to Boss in Mazinkaiser, when the Gamia Q sisters attempt to assassinate him, thinking he's Kouji. His school uniform and his boxers are torn to ribbons by their hair. 
  


Doesn't end there, by Cutie Honey, he's gotten quite good at it: