This exact setup happens all the time in movies, so let's examine how they handle it.
If they are the only character, then as a GM, completely tune the story to them. They shouldn't have to do hacking, or at best they have to shoot their way in to where the Russian hacker who already knows stuff is. James Bond doesn't use keyboards. Avoid his minimums except for once in a while for dramatic effect, or to introduce Bond Girls who have that skill set. As for addressing his strength - he's an expert shooter, but is always having to go places where he doesn't have a gun, or just has a pistol when others have machine guns, or has a machine gun when they have tanks. Or places where you can't just shoot the heck out of everyone (like a public casino) without having long term consequences.
If they are not the only character, the other characters are in danger. What do The Killer and Hard Boiled have in common? Lots of OTHER people who get killed. The chick who's along that's in danger. Or your buddy movies with one killer type and one intellectual (or even just not a killer, take Rush Hour) - the killer has to spend a lot of their time protecting/coaching/handling the less combat oriented person. One of the big risks of having a min-maxed PC in the group is the min-maxed bad guys the GM has to toss on, who can often terminate the non min-maxed PCs in a round. It becomes the combat monster PC's job to avoid that, or else the whole party dies and they say "new game, and be less of a goon this time please."
This is of course advice for in-play. You should try to head this off ahead of time by disallowing (GM)/forgoing (player) total min-maxing by choice of system or GM guidance. Because as you note it ends up being unsatisfying even in your maxed area.
As you state correctly, a character can have a maximum number of skill ranks in any skill equal to its Hit Dice.
Hit Die, singular, (i.e. the bards d8) is a basic component of a class and only determines the amount of hit points a character gets upon taking a level in the class1. It is (hopefully) never abbreviated HD.
Hit Dice (or HD), plural, means the number of hit dice you have, not the size of the die itself. For regular characters (e.g. standard races), this is equal to the total amount of class levels he has, i.e. his character level.
Monsters also have Racial Hit Dice, which are added to all class levels they possess (if any) to determine their Hit Dice2.
In statblock notation (like in the Bestiaries), it is noted as XdY+C, where X is the Hit Dice, Y the Hit Die, and C all constants (CON-Bonus, Toughness, etc.). If there are more than one type of Hit Die, there are more terms like the first one. The total Hit Dice of the creature then is the sum of all Xs.
Some quick Examples:
- Human Bard 4: 4 HD: all d8.
4d8+C
- Half-Orc Barbarian 4/Wizard 8: 12 HD, 4 d12 (Barbarian) and 8 d6 (Wizard).
4d12+8d6+C
- Wolf Ranger 7: 9 HD: 2 d8 (Wolf/Animal), 7 d10 (Ranger).
2d8+7d10+C
1 The hit die also usually determines the BAB progression of the class. This holds true for all Paizo classes as of this moment, but not all 3PP classes follow this trend.
2 For a more thourough explanation, see Jack Lesnie's Answer
Best Answer
First decide IF the check can succeed
In your example of a Griffon, they have the following text in their decriptions:
[Emphasis mine]
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:
Nature This sounds like it has potential uses, but the PHB suggests this is to recall lore about nature, so this is out.
Survival 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.
Persuasion Maybe you can talk to a monstrosity? Well this is for talking to people specifically, so again, by the PHB this is out.
Animal handling 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.