There is a pet, which I have in my alien containment smile

I'll spoiler it

I believe you are talking about the Cuddlefish. It can respond to commands, and if released in the 'wild' will follow the player around, until it either disappears or dies a gruesome death. Because I don't want the cute little things to die, I keep mine in alien containment in my Deep Grand Reef base. I have two in there, and more eggs in storage

If you have picked up any creature eggs from one of the Degasi bases they are Cuddlefish eggs. I think I also found them in alien structures

From the wiki:

The Cuddlefish will either follow the player around or stay in one area when told to. The player can tell it to stop/start following them using RMB.

The player can play with it using LMB. It has various play actions, such as high fives, magic tricks, or cuddles, and will display various emotions depending on said action.

That's a fine looking Seamoth Coot.

Animals flee this hell, the hardest stones cannot bear it for long. Only men endure