Axolotls were first announced on Minecraft Live 2020 and released at Minecraft 1.17, and they quickly became very popular in the Minecraft community, in case you do not know, they raise awareness over the real-life amphibian called Axolotl, or Ambystoma mexicanum, so yes, they are amphibians and not fishes!
The Axolotls were near extinction from 2020, they are critically endangered and have a decreasing population, even though these little creatures are extensively researched due to their capacity of regenerating limbs. If you want to know more about the real-life Axolotls, take a look at the Axolotl Wikipedia page, and you can also know more about their population decline on National Geographic.
Photograph by Anna Mizi, National Geographic Your Shot. Image source
As for Minecraft, Axolotls are a great friend to keep around, they're easy to tame and they'll protect you from a lot of danger. Axolotls are passive towards the player, but they're aggressive towards other underwater creatures, the same way a golem will protect the village, Axolotls will protect their habitat.
To tame an Axolotl all you need is a water bucket! By using a water bucket, you're able to carry the Axolotl with you and bring them next to your home or hold them with you until you need help with underwater creatures. The Axolotl, however, won't be following you like a wolf or a cat do, they can be carried around and breed, but it is uncertain if they will have the same features as the other pets, which is the actual taming in Minecraft.
If you want to breed an Axolotl, you need a bucket of tropical fish, you can get this item by carrying a fish in a water bucket, the same way you do to get an Axolotl. The bucket holds a still alive mob while throwing a fish that isn't inside a bucket would be a fish that is already dead, and Axolotls prefer living prey!
Recently with Minecraft 1.17.1, captured Axolotls won't attempt to leave water constantly, this happens with small bodies of water, as related in MC-222731, it won't be a big problem in most situations, but it's a good patch. Another change from 1.17.1 is that blue Axolotl will only spawn through breeding, so you won't be finding them on caves, if you like blue Axolotls be prepared to bring a lot of tropical fishes!
Where I can find Axolotl in Minecraft?
You know how to tame them and how their breeding works, but where to find them? Axolotl won't be found in the ocean like other underwater mobs, to find an Axolotl, first, you need to find a Lush Cave(not really for now).
Lush Cave is an underground biome, covered with moss, lit up by Glow Berries, these caves look different than the regular caves and you'll notice that something is different due to their unique appearance.
Sadly, the Lush Cave biome is only available in a Buffet or Custom world type for the Java Edition, for Bedrock edition through the experimental feature option. They will probably be fully accessible further on, with more testing and development, it is expected to be on the 1.18 update, for now, you will find Axolotls in underwater areas inside caves.
Further, Axolotls will only be found inside the Lush Cave biome, and you're able to find these biomes by spotting an Azalea tree, this is a tree that has its roots inside Lush Caves, so they will only grow above Lush Caves, once you find them, look for an entrance or mine down.
After you team up with an Axolotl in a fight you will get the "The Healing Power of Friendship!" achievement, enjoy your new amphibian friend!