The in-game unlock requirement given in game is as follows:
Own a penguin and survive a week
I've never seen anyone online mention how to complete this. The guide by warfreak states:
(This is a major problem as many people have emailed and complained that they had a penguin and survived a week on it and no medal. I have no idea either. Ask Maxis or EA for some help)
If anyone has unlocked it, they haven't shared that information online, and searching the topic online only brings up old forum posts from players asking how to unlock it. But I know. I know how to unlock it. (the only comment online mentioning how to complete it was also written by me)
To unlock the Tuxedo medal, the penguin, purchased from Unsinkable Treasures for §15000, must be your only pet. Your dog or cat must not be in your house at the time.
You may board your pet at the kennel, as long as you have no other pets except the penguin placed in your house. The quest can also be obtained before obtaining your first dog or pet, but putting that quest off in order to save up §15000 just to buy this penguin, locking yourself out of progression just for this achievement is at least Slightly unhinged.
As for "surviving a week," I didn't run multiple tests as passing one in-game week takes a bit more time than I have patience for, let alone repeating it, but for that week, I stayed in my house, surviving off the amenities I already had.