The cake is a lie.
Spoken like a true Dragon Age fan.
I've played TS3 on both PS3 and PC, and the birthday cake procedure is pretty much the same on both.
I usually buy a cake from build/buy mode instead of going to the store to purchase one. You can easily place them on the table or a counter, just like any table lamp, small appliance, etc. Birthday cakes need to be sitting on a table or counter-top to be used to trigger the age transition. When you're ready to commence the birthday celebration, select the birthday Sim and then click on the cake and choose the "blow out candles" option (on PS3, you need to have the birthday Sim selected when you interact with the cake; the PC version gives you a pop-up list from which to select the Sim that is to age up).
After you've done that, the rest of it happens automatically. If the birthday Sim is a baby, an adult will carry the child to the cake. Other Sims present will blow party horns, etc., and the birthday Sim will blow out the candles and age up. Then they eat the cake. Or sometimes, they just take a piece, set it on the floor, and bail. When that happens, you'll have more leftovers.
I've never tried putting a whole cake in the fridge - but I'm guessing that once you do that, it is treated like any other food leftovers and no longer has its "trigger age transition" mojo.
Good luck.