Nor should it, as the two are unrelated.
That is absolutely the wrong approach.
If your NAS is running DSM 6, then I’m surprised Plex is still working correctly.
If you NAS is running DSM 7, then what you did had zero effect, as Plex Media Server does not use the Plex username on DSM 7 based systems.
Changing your Plex password unclaims the server (removes it from your account).
Reclaiming a server can be accomplished by several methods.
The easiest, if you are local to the NAS, is to point a browser at its IP address, http://syno_ip_address:32400/web.
If that does not work, on systems running DSM 7, Plex can be uninstalled then re-installed using a claim token. The process is outlined in FAQ 31: Reset / Reclaim PMS on DSM 7 (Not Authorized - Locked Out).
For NAS running DSM 6, follow FAQ 15: Claiming & Connecting to a troublesome Synology server.
