When you activate 'The Elder Scrolls V: Skyrim Anniversary Edition' you will see the base game of the Special Edition is provided.
This is correct-- Actually, you do have the Anniversary edition, the additional DLC is applied to the base Special edition game.
How to check in Steam
If you have the Steam edition of this game please right click on the Special Edition and choose Properties, then choose DLC.
You will see Anniversary DLC is listed.
How to check in GoG
Method 1
From the GoG Galaxy Windows application, log in to the account you activated the game on and choose Orders History.
Observe the Anniversary edition.
When you install the game, 'Special edition' will be installed with the extra Anniversary DLC.
Method 2
One way to confirm is to visit your Gog account at
Select 'Download Offline Backup Game Installer' and observe the two Anniversary DLC downloads