Minecraft Legends

You have to pet 10 (11?) different animals. You can't pet mounts, so those are not mounts!
Most of the animals you can find near the Well of Fate (start spawn).
To pet the animal you have to come close to it and long press , there will be hearts around it.

List of the animals:
1. Wolf
2. Chicken
3. Llama
4. Ocelot
5. Pig
6. Rabbit
7. Marmot
8. Badger
9. Turtle
10. Cat - I believe you can find them only in the villages
11. Fox - thanks to @Ohhh Declan !

I can also confirm that this achievement can be unlocked in Custom game

Edit: I was lacking 1 pet in the end, it could be a bug of progress counter. But after I pet all 10 animals several times I found a cat and it popped. Cheers!

This achievement can be done very quickly on a new world with on Story mode difficulty.
Launch a story mode campaign and skip the tutorial.
At the well of fates you'll find 10 out of 11 needed being:
1. Wolf
2. Chicken
3. Llama
4. Ocelot
5. Pig
6. Rabbit
7. Marmot
8. Badger
9. Turtle
10. Fox

11. Cat, to get the last one for the achievement go defend the village, I used peashooters to make it go as quick as possible and there will be multiple cats available to pet.
Congrats, you should now have the achievement finished!

I tried with my mate multiple times on multiple worlds in a custom game and couldn't get any cats to spawn. I was able to get cats to spawn in a custom game when I hadn't changed any settings but I couldn't find out which setting makes cats not spawn. To play it safe just do a normal story mode campaign to guarantee the spawn.

First make a new world ideally in customs. After some cutsceens you spawn in. At the base spawn you should be able to get 10/11 pets to pet which are: badger, chicken, fox, llama, marmot, ocelot, pig, rabbit, turtle, and wolf.
You may have to do a couple of circles around the well of fate. But those base 10 should spawn in. If not try walking away a bit and return to the well.

The cat is the last pet and will be 11/11.
They are found only in villages and can be annoying to spawn. Just teleport to one of the villages surrounding the well. If the cat doesn't spawn in try walking away from the village and come back. I just so happened to have almost all the cats spawn in a singular tree at the village.

Something to note*
If you beat the game your well of fate will be damaged as well as the surrounding area. I noticed that the animals no longer spawned around it. In this case its best to make a new game again ideally in customs to be quick.