You don’t setup a new server if moving your Synology based Plex server.
Do you have a Synology package or Docker container install and what OS are you moving Plex server to?
If you have a package install of Plex on your Synology NAS, this is the sequence of events from memory, and I’ve moved plenty of test servers myself.
In the webapp of your current Synology based server, turn off automatic library scanning and scheduled scans, while also disabling automatically emptying the trash when scanning libraries.
Go to troubleshooting in the webapp and for good measure, and clean bundles and optimize your database.
Then go into authorized Devices and switch to the server option to remove the reference to your Plex server from your list.
This does not destroy your Plex server and only means it needs to be reclaimed, which you won’t do until it’s transfered and running on the new hardware.
Shut down that instance of Plex server running on your Synology NAS.
In Synology DSM and their File Station applet, find the PlexMediaServer shred folder and navigate down to two levels to see the Plex Media Server subfolder along with a tmp subfolder.
Right click on the Plex Media Server folder and pick Compress to Plex Media Folder zip.
Pop that onto a good flash drive or transfer it across your local network to the new server hardware.
Install Plex on the new system but don’t start it, or stop it if it automatically starts.
Without knowing what your new OS is, I can’t guide you to the correct subfolder, but Plex has a supoort article on both application folder locations and a support article on how to move your server.
Find the corresponding Plex Media Server subfolder on the new install and delete it to replace it with your zipped contents from your old server.
Start the server up and in any local web browser, as long as you haven’t setup multiple subnets, find the transplanted server with this address once you’ve put in your own local IP address to the new server location.
http://server.local.ip.address:32400/web
There’s another Support Article called Opening Plex Web App that has that information.
Claim your server and hopefully, you’ll see the old server name while doing so.
You’ll have to setup a new port forwarding rule in your router’s settings and ditch the old rule if your not using UPnP, and in either case you’ll have to revisit the Remote Access settings of the Webapp to gain remote access again.
Again, this is all from memory so please visit the relevant Plex supoort articles I’ve mentioned.