The fonts that you licensed years ago from Adobe are no longer directly licensed by Adobe except as part of the (very expensive) Adobe Font Folio product. The Adobe Fonts service, available with licenses to the Adobe Creative Cloud and Document Cloud products (which temporarily install access to fonts), does not contain most of the fonts of the older Adobe Font Library (i.e., the fonts still available on the Adobe Font Folio).
Given that your original purchase from Adobe was a perpetual license to use those fonts, you certainly may install the corresponding font files on your new computer.
I think that the problem that you seem to have is that you expect to be able to download those fonts again from Adobe. That is what is not possible anymore. Adobe is under no obligation to indefinitely provide “backup copies” of software products that you licensed long ago; this is common industry practice.
What I would recommend is that you retrieve those font files from your backups of your old computer that had those fonts installed and install the fonts from those files on your new computer system. That would solve the problem and is perfectly kosher! You do backup your computer regularly and keep those backups, correct?