Edit: If you’re using dev mode, you can use the command item cutefishegg to spawn an egg into your inventory, though as with the other achievements, you’ll need to save, quit and reload your save in order to unlock the achievement.
In addition to NerdsLastStand’s guide, this achievement will unlock when you “play” with your cuddlefish. There will be a brief, sweet animation where you high-5 your new buddy when you interact with it.
This may just be a bug in the new build (an update came out a few days ago), but I personally couldn’t interact with it whilst we were inside the Alien Containment module (the aquarium in your base).
If this happens to you simply pick up the fish into your inventory, go outside your base into the main waters and drop the fish from your inventory. It should now let you interact and play with it (though note it won’t let you pick it up again, so if you want to keep it in your base, have a save before doing this)