I've never heard that phrase before! It sounds like a Chinese proverb to me, so I looked it up and got this explanation:

"There isn't a specific Chinese proverb that directly translates to "you have a key and a cake in your hand," but the closest idiom that conveys a similar idea of having both the means and the reward is "得鱼忘筌" (dé yú wàng qián), which means "forget the net after catching the fish" or "having the cake and forgetting the key"."

If I've understood it right, to "have the cake and forget the key" is like as soon as you've got the prize or reward, you immediately forgot the tool that helped you do that. Basically, you took for granted the fact that without the net, you would have no fish, or without the key to unlock the box, you wouldn't have got the cake that was inside.

In the case of your fortune cookie, it seems to be either a mistranslation, or they've invented something new, like, "You have the cake, but don't forget that the key was what allowed you to have the cake," but that last part is a bit of interpretation on my part!