This achievement is unlocked by having the dog companion, Pancake, in your shelter as the cat companion Sharikov joins you.
You have to get Pancake first. You can not get him after Sharikov entersThe first step to unlocking this achievement is getting the companion Pancake to join your shelter. Doing this will unlock this other achievement as well:
To get pancake to enter your bunker, you will need (1) A flashlight (2) One can of food (3) A map/Axe (4) A med kit.
The first pop up will mention growling and glowing eyes. Just make sure to use the flashlight here to not kill Pancake. The rest of his missions will mention specifically that it is for him.
After Pancake is with you, you can get Sharikov by trading a man that has a bag a Soup Can or other pop ups that mention allowing a cat to enter your bunker
Once you get Sharikov, he will end up forcing Pancake out. This will immediately unlock the achievement.