Is Plex library synchronization automatic or does it always remain manual? I added some tracks on Plex but to see them in Yatse I had to update manually.
Here the log
debug.log (9.4 MB)
It’s automatic when you browse but throttled to 15 min minimal check interval.
Your logs does not show a test for library change so the last one was less than 15 min earlier.
OK thanks
but is this also true when the app is in background?
Well no since I said when you browse 
There’s no need to sync every 15 minutes and consume battery when you can only have 1 sync at the moment you need.
If you want background sync Yatse have an API that can be used from Tasker for example to sync when you prefer.