Thank you Sbcroix for your research. I've concluded the same thing. Adam.storrow also has the same problems, so in other words, this is a new Plex Media Server update problem that happens to PS4 client users, therefore it's up to Plex to handle it.
Older Plex versions worked with my PS4 for over 1 year without any problems with subtitles for me and my PS4. For the moment I've rolled back to the old Plex Media Server (the version before the latest), and that works okay. I still get the error from time to time, but usually only one time per session, the first time I try to start a video. With the latest version, I can't start any video no matter how many times I try.