I will never tame one of these. I mean to tame a level 1, you will need 439 cooked prime meat and it will take 19 and a half hours! You will also need 1310 biotoxin to keep it knocked out. And a level 150 will take 149,42 hours to tame if you're using cooked prime meat! Such a waste of time and resources. Up if you want wildcard to make the royal griffin easier to tame.
To all who are wanting to tame one: (WARNING! WALL OF TEXT)
If you are level 40+, your surname needs to match that of one of these creatures. For example, if your name in the game is Sasha, but the one you want to tame has the name Morphi’i, you will need to change your name with a name change ticket (100 amber) to Sashai’i. If you are level 95+ or have the gold crown skin on your helmet, then you can disregard needing to change your name.
Next, a trap of a 4x4x4 cube with 4 wooden Dino gates is the best way to trap it. Griffins can damage wood, but I was able to trap & tame 2 with a single of the structure aforementioned. When taming, it is important to note that if you do not have a soothing balm, it is going to take literal hours to tame the griffin. A x10 taming balm is decent, and easily obtained from a dungeon (the southern one near the center is my favored one for gathering these), but a x15 is preferable. Keep in mind even with a x15 it will still probably take 20+ minutes to tame *at best.*
For food, it only accepts prime meat, prime fish meat, either raw or cooked (but raw is better, of course), or allosaurus kibble. To get a griffin to lay eggs, you must have the female mate with a male, and griffins only mate with another griffin of the same surname. Go for a low level, always, unless you are prepared with thousands of narcotics and armed with the patience of a saint.
Once tamed, lead your griffin back home and put it in a safe space to let it passively level up. Once it’s health is at least x3 that of a normal griffin and it’s stamina is at least x2 of its original value, you’re ready to take it out on flights.
Furthermore, breeding griffins will produce better offspring, just like with all other dinos, but again, be armed with the patience of a saint. It takes many real-life days, about 2 weeks is my guess, of nonstop gameplay, for the baby to mature completely. While it is in its baby phase, it must be near its mother to feel secure enough to eat. However, once it is juvenile, it can be moved away from its mother if necessary. Hatching the egg will take at least 5 air conditioners.
TL;DR: Royal griffins are mainly trophy tames, but once trained enough, they are a cross between argentavis for firepower, and tapejaras for flight mobility, with a large aoe attack.