First decide IF the check can succeed
In your example of a Griffon, they have the following text in their decriptions:
Once trained, a griffon is a fierce and loyal steed. It bonds with one master for life, fighting to the death to protect that rider.
This suggests that a second person would not be able to really interact with even a trained griffon mount. These are not horses which will take any rider.
There are more monstrosities than any answer here can go through, and some may be tameable like a horse, but generally short of being the one that raised one from birth it is unlikely you can succeed at all.
I can’t actually find the rule that says as much, but if there is no chance of success (or no interesting consequence of failure), don’t roll.
But as a DM I have decided there is a chance of success in my world
As a DM you are free to override the default assumptions so maybe in your world a griffon will submit to anyone once it has been trained, and you need to work out how to do this.
Look at the list of skills and see if any might be useful. I am not going to list them all here, but the few that might be useful (in my opinion) are as follows:
This sounds like it has potential uses, but the PHB suggests this is to recall lore about nature, so this is out.
The PHB suggests this can be used to find signs of an owlbear, so clearly has some interaction with monstrosities, but only tracking, so this is out.
Maybe you can talk to a monstrosity? Well this is for talking to people specifically, so again, by the PHB this is out.
And this is where I have to disagree with the linked answer. The logic there is that a beast being an ordinary animal means that nothing else can be an animal, but that isn’t correct. It leaves a whole classification of extraordinary animals to be classified, and those, at least some of those, could well be monstrosities.
If you rule out animal handling as being a useful skill here, then effectively you are saying there is no useful skill, but someone tamed that griffon in the first place, so there must be at least one useful skill. The trick isn’t to try and rule every skill out based on a strict reading, but to find the most closely related skill – and that is animal handling.