Make a copy you control
Open My List and choose Create backup. Copy all the text from the resulting field into your own note or document. It contains saved references and valid checkpoints, not videos or streaming-account passwords. Treat it as a personal list of viewing interests.
Merge rather than start again
On the other browser, open the same MusicHQ site and paste the full text into Restore backup. Choose Merge backup and read the result. Valid new titles can be added while existing checkpoints in that browser remain. An unknown title or an invalid episode cannot create a new catalog entry.
If the text is rejected
Check that the entire backup was copied without editing its punctuation. A backup from another brand is not a MusicHQ backup. Keep the original until your restored list looks right. This is a deliberate transfer you make, not automatic account synchronization between all your devices.