The only ice cream cake I’ve ever seen (and I’ve seen a lot, my in-laws love the stuff) has a layer of cookie crumbles inside. I’ve never seen a single one that has any sort of cake in it whatsoever. I also did a google before posting my comment and not a single recipe I found had cake in it either. So I’m very sorry, but no, ice cream cake does not (typically) have cake in it.
It's not that fussy, you just need the freezer space and a plan. Slightly softened ice cream is easy to layer on and picking the right frosting keeps it from cracking. (I used a sweet marscapone) Totally worth it, btw.