Moving to a new device
Keeping the content
When moving to a new device, you can take your library with you.
As explained here, Panels creates a new library inside Files -> On my iPhone/iPad -> Panels.
Panels device folder does not get backed up with your iPhone backup.
Before deleting all content on your current iOS device, make sure to save a copy of the Panels folder somewhere (in your mac, external drive, etc.). You can also airdrop the whole folder to your new device.
Once you have your new device, install Panels and go to Files app. Go to Files -> On my iPhone/iPad -> Panels and drop all the content from your other device in there.
Once you open Panels, it will start reading and indexing its content.
If you are using iCloud Drive or OPDS, you don't have to do anything other than just setting up the library in your new device.
Keeping your reading sessions
There is no way to "send" your reading sessions to a new device manually. The only way is to log in to Panels (via any of the sign in methods) in your old device (if you are not already) and wait for all the reading sessions to be uploaded.
On your new device, log in to the same account and the new device will start downloading all sessions.