Can I get sick from eating 3-month-old ice cream cake?

You would be surprised at how much your mind will factor into this equation. If you aren't sure about it 100 percent in your mind you will run a greater chance of FEELING ill.
As stated in another contributors answer, Bacteria growth slows to a near stop when frozen. This being an ice cream cake you should be more than fine in all physical aspects.
Mentally you are on your own lol.

Of course not. Why would you think that frozen food would go bad? This is why we invented freezers, to keep the food good for the future.

so why do you think the cake would spoil badly enough to make you sick while in the freezer?
If that would happen there would be no point to freezing anything.
there is nothing wrong with it.

If it's been in the freezer, and the freezer hasn't lost power, the ice cream and cake should be fine, maybe a bit stale/dried out, but otherwise fine.

yep