So to my understanding you ALLWAYS have 1 Ice Key, however you never see that you do have it until a flag data tells the game to show that you do indeed have one,(seeing as it would be easier programming wise to just always let you have one and only let you use it after its revealed).
Now I have two possibly dumb questions but:
1: Does it affect the game clear % at all? I know it probably doesn’t because you can start a new game immediately save and quit and it will still show 0 %.
2: What if you forced the game to not give you and Ice Key until you actually start playing? It might show up if it's collected while the game is playing and not just in a File select screen.
This is just a guess but thanks for your time.