According to the post on Minecraft.net, the patch notes, the Minecraft Wiki, and every Minecraft player, axolotls are tamed by putting them in a bucket. However, they don't actually become tamed. They don't follow you around, sit when you press the use(defaults to right click on most mice) button, or even have any NBT data that is shared with other tamable mobs. On top of all this, if you're stupid or heartless enough to kill these adorable creatures, they display no death message no matter how you spawned them. Also, releasing one from a bucket doesn't grant the "Best Friends Forever" advancement.