My best guess is the idea is that it maps to aspects of a user story like "get input from the user," "do some business logic," "show output to user." So even the first slice is a working prototype in some superficial sense. The elephant organs might be app components (UI, database, etc.), but in the first slice you don't have a complete UI (maybe you have text input) and you don't have a production database (maybe you just have an in-memory dictionary) and you don't have robust business logic. You have the whole stack, but each part of the stack is incomplete. That's what I think makes it a vertical slice.
A horizontal slice (what not to do) would be one complete elephant organ. Maybe that's a production transactional database. So in the first slice you have a complete database or you've written the final business logic, but none of the other things that you would need in a mockup/prototype/MVP or an integration test.
Anyway, this is my best guess.
I found the comments really helpful so I wrote up some thoughts on different approaches to tackling projects. Hope this might be helpful to others.
https://joshuatauberer.medium.com/bullet-time-and-elephant-h...