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.
If you have encounter the bug preventing you from interacting with it whilst inside the Alien Containment module (the aquarium in your base), the following helped me:
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)