There's no legal requirement [in the USA] that the CC0 license and authorship information be retained. The only risk that I know of for claiming someone else's CC0 material as your own is that you might be publicly shamed for it.
As an experiment, I ("Alice") released some code under CC0 on Stack Overflow several months ago.
Here's the license as I applied it:
/* rgbtobgr565 - convert 24-bit RGB pixels to 16-bit BGR565 pixels
Written in 2016 by [Alice < firstname.lastname@example.org >]
To the extent possible under law, the author has dedicated all
copyright and related and neighboring rights to this software to the
public domain worldwide. This software is distributed without any
warranty. See http://creativecommons.org/publicdomain/zero/1.0/.
Then I simply waited to see what would happen. Until now, nothing happened. Today, I found that the same code just with some minor changes and with all the CC0 license information removed and replaced with a GPL license and a "Written by [Bob]" line that claims authorship by another individual.
Now, I don't see a problem with the relicensing nor with removing the attribution. But claiming authorship is problematic. In some jurisdictions and in most academic circles, that's frowned upon.
I suppose I should not be surprised how it has turned out so far.
FOLLOWUP: After a little discussion, this particular case ended satisfactorily. "Bob" has added a one-line attribution, not legally required but a nice thing to do, within the body of his code:
// This part was under CC0 Licenses and was written by [Alice]