Mark of Handling
The Mark of Handling gives its bearer a primal connection to beasts and the natural world, granting the power to calm and coax. This extends beyond purely natural animals; the mark allows its bearer to guide a hippogriff as easily as a horse.
Wild Intuition. Whenever you make a Wisdom (Animal Handling) or a Intelligence (Nature) check, you can roll a d4 and add the number rolled to the total ability check.
The Bigger They Are. Starting at 3rd level, you can target a Beast or a Monstrosity when you cast Animal Friendship or Speak With Animals, provided that the creature's intelligence is 3 or lower.
I apologize for starting off with a block quote, but a search of previously asked questions didn't provide links to anything immediately obvious that suggested this subrace in particular has even been addressed here, so if I'm going to seek a subjective answer, I thought it best to open up with context and demonstrated research, and in this case that requires citation.
So this is a pretty deep well. After thoroughly reading these (related) questions, and the associated answers, one particular comment by Darth Pseudonym stands out to me as particularly insightful and relevant here.
The skill uses given in the book are not the be-all and end-all of what the skill can do; they're "examples of how to use" skills (PHB p.174). It's improper to claim a skill can't do something if it isn't listed in the book and therefore a 20 on the roll does nothing, or should not be allowed. Rather, say the book does not specify the results if a DM allows a non-standard skill usage, which means it's up to the DM to determine how to adjudicate it.
I agree wholeheartedly with this sentiment, but I'm sitting in the player's seat for this one.
The idea is a Mark of Handling Moon druid who has a personality trait of preferring the company of wild creatures to most people- "People are dishonest and hide their nature, while wild creatures embrace it for all the world to see," and a flaw of "All the wild creatures I meet are potential allies- ALL OF THEM."
I'm not planning to go out and track down the Tarrasque or something blatantly game-breaking and insane like that. However, I find it entirely plausible that I may at some point come across things like griffons, hippogriffs, pegasi, unicorns, owlbears, worgs, gorgons, rust monsters, cockatrices, or even rocs.
All of these creatures fit the classical archetype of 'wild animals' with the exception of the fact that they're magical fantasy creatures - 'monstrosities.' Numerous of these creatures would make welcome companions to the party. The druid in question is invested heavily in both wisdom and charisma, and so to speak, would like to eventually 'catch them all,' in-so-far as the opportunity presents itself without derailing from the actual mission at the table.
Because this is an off-text application of related skills, and because there's as much a social component as a 'handling' component when approached this way, the druid is investing ASI levels into both wisdom and charisma through multiple skill expertise feats, gaining expertise in animal handling, athletics, deception, intimidation, insight, perception, and persuasion (5 Skill Expertise + 1 rogue level at level 1.) This ends with 20 wisdom and 18 charisma.
The point of all this is that I've gone for covering the bases of the most likely type of 'social' interaction with monstrosities with expertise, from racing and wrestling, to posturing and growling, dealing with them honestly or telling them a lie (come hunt with us, we find the tastiest meats/we didn't steal your egg, it was that harpy that just flew over the mountain behind you). My approach here is limited only by the situation and my creativity.
Please, Fallen, my eyes are bleeding; what exactly are you asking?
What I'm looking for is suggestions on ways I can make this fun for the other people at the table with me. There's an easy-bake happy button for the party in being able to potentially give tame flying/fantastic mounts/guard pets for them- and the idea sounds fun in my head. But I don't want my pokemon stable to dominate the entire play session, and I don't want to make the DM's job so difficult they have to give consideration to just cancelling or re-rolling any random encounters involving wild monstrosities.
Are there any established rules or costs to taming/training such creatures? If not, has anyone come up with a system of checks utilizing downtime costs or other measures to keep this from becoming a major problem on either side of the table? If so, how did it work for you?