You can tame up these mostly raid used tames as the following ways:

Lvl40: You need to buy a name change ticket and chante your surname same as you choosen Griffin. (F.e. Griffin: ThégaRIC, Your: PlayeRIC)

Lvl60: Buy the Gold Crown (Primal Pass required) and so you can tame up any.

Lvl85: This is the first level number, when you can tame up several Griffin, but not the high level ones.

Lvl90: You can tame up any Griffin you want.

Tips and tactics: Use the classical Griffin trap and use Darts, bring Bio-Toxin which is cheap and useful, use Allosaurus kibble and 10-15x Balm or else it will take hours.