This can be done with several factions, as long as the lords can have mounts. So Dwarves can be ruled out
I got this achievement playing as the Elves. Elves have Glade Lords (Elven lord) that can have these "special" mounts. Level up the Glade Lord by conquering enemy settlements and fighting battles against enemy armies.
As soon as the Lord is level 10, go to the skill screen. At the top is a bar with several unlocks, including mounts. Choose the skill that gives your lord an Eagle as a mount and the achievement should unlock. This should also be doable with a Forest Dragon mount, but that requires the Glade Lord to be level 19.
Another option would be the Chaos Warriors. Chaos Lords can have a Chaos Dragon or Manticore as a mount.
Playing The Empire you will be able to get pegasus and griffin mounts. And as the Crooked Moon you can get Wolf and Spider mounts.
I will update the solution when I encounter more races with such mounts.