After some time and many people asking Crawford about it, he finally gave us his word. And well, it doesn't help much, but now you have something to show your DM. The first twitter is here.
Want your D&D character to have a pet/companion?
Here’s a little secret: you don’t need special rules for this. Through roleplaying and ability checks (most likely Animal Handling or Persuasion), you can have a buddy, as long as your DM is OK adding a creature to the group. #DnD
There was a chain of discussion on his twitter about it, which I will post the most relevants to the discussion
The default assumption in the D&D rules is that you can befriend people and critters you meet on your adventures.
If a class has a special companion (Beast Master, find familiar, find steed, animate dead, etc.), that companion is in addition to creatures you might befriend. #DnD Source
When asked about leveling up (increasing the companion's HP and everything else), he cites the DMG rules on that.
The “Dungeon Master’s Guide” has guidelines for adding class levels to creatures.
Say your wolf friend has been fighting by your side for an adventure, the DM might give the wolf a level in fighter. This is a legitimate use of the DMG rules. #DnD
Want to add class levels to a monster or NPC? Take a look at the section "Monsters with Classes" in the "Dungeon Master's Guide" (p. 283). #DnD Source
He further argues about the "letting other people get animal companions make the beast master useless" later
There are various ways to gain thematic companions in D&D. Here are some of them:
• spells like animal friendship, animate dead, find familiar, and find steed
• the Beast Master
Each way gives a different experience. Choose the one that best fits your story! #DnD Source
So, officially, it's about role-playing accordingly. First you would need to befriend the creature. Then, if you want him to listen to your commands, some kind of downtime training, to be decided by your DM, or making persuasion checks every time. The more friendly you are with the creature, the easier they should be, though. Finally, just use the usual NPC rules for having the companion in the party, giving him experience and even leveling him up.