> If someone else uses your exact same prompt to generate the exact same code, can you claim copyright infringement against them?

No, because they've independently obtained it from the same source that you did, so their copy is "upstream" of your imposing of a new license.

Realistically, adding a license to public domain work is only really meaningful when you've used it as a starting point for something else, and want to apply your license to the derivative work.

Copyright infringement is triggered by the act of copying, not by having the same bytes.