Time machine: moving a source external disk to a new computer
To make sure I understand - you have one external HD for data, plugged into your new computer, that was previously attached to your OLD computer - and you backed up your old computer + this drive to a second Time Machine drive?

Time Machine is not recognizing the old archive because the new machine is a fresh install. So it made a new backup. The data drive (which you state is being seen as a "new drive") would have been part of that old machine backup - Time Machine should recognize things on a per-computer, not per-drive basis. Unless you tell your new machine to inherit the old backup, the existing backup of your data disk will be ignored as well, and a new backup will be created with that disk as part of your current computer backup.

SSDs are great but the lack of cheap internal storage means more people have to use external disks for data. Cloud isn't a good alternative for everyone. Unfortunately the backup systems designed by Apple are not treating externals as separate backups - they are included as part of the internal mounted disk hierarchy of your computer.