The Thalmor know Talos is a god. They just hate him. Banning his worship was done mainly for two reasons:
1) It drives a wedge into the Empire that weakens it from the inside.
2) Humans believe that after the beginning of creation, the gods brought about the existence of mortals from the earth bones. This makes them revere Lorkhan for starting creation. Elves believe that they descended from the first beings to form and existed as spirits prior to creation. They hate Lorkhan for starting creation and forcing their spirits into a cycle of life and death as mortals.
Talos, having taken Lorkhan's mantle in the Wheel of Convention, is thus to be hated in their eyes. The Thalmor's ultimate goal is to finish destroying or disabling the remaining towers that hold together the Wheel, thus ripping apart creation and returning the universe to how it was before Lorkhan's betrayal tricked the gods into creation.
He's a divine.
But he's not an aedra.
Talos is Lorkhan. He may not be considered aedric because I don't think Lorkhan is considered aedric. The deity that tricked the gods into creation didn't technically participate in it.