The penguin is a seemingly normal stuffed penguin found in the apartments in the Refuge. It serves no purpose to the overall plot. If Niko interacts with the penguin and says "Hello, penguin!" they are greeted by an audible, raspy voice saying "hello."
This is a homage to a joke game Hello Penguin developed by The Hello Penguin Team (Noxid, PluswPlus, Brabbit and HyMyNameIsMatt), with Noxid who voiced the penguin plush.
There is a secret room accessible from the clock room in the Refuge. If "HELLO PENGUIN" is entered into the keypad on a run that isn't Solstice or the first run, a secret penguin room will open showing a much bigger version of the penguin. This can only be done once per run.