I am added as an admin and user to the Adobe Target Suite but I still do not see the card allowing me to access the Adobe Target Classic.
Target Classic has been decommissioned some time ago. Please share the Target Standard/Premium permissions with your account and you should be able to see a Target Card here to access Target Standard/Premium
Target Classic has been decommissioned, please go to admin console Admin Console and add Target Standard/Premium to your profile on Enterprise Cloud.
Hope that helps.