How do I add an eReader to my Adobe ID?

Most tablets do eReader function via a standalone app such as Bluefire, not via ADE (digtial editions).

Try installing Bluefire and then registering that with your Adobe ID.

If you need to copy files from your computer/ADE, use a file manager (eg Windows Explorer, Mac Finder) to copy to the tablet as a mounted drive,

and then open them in Bluefire.

If the tablet is an Apple one thye make it more difficult to transfer,

you can't use file manager but must use Dropbox, iTunes, eMail yourself or similar to transfer the files.