Hopefully, all you'd need to do is to repack the ZIP so that the files are at the ZIP's root instead of inside a single main folder holding everything.* I'm not sure either, since my uploaded package did it right off the bat and I didn't need to do anything special. It may also** have something to do with the main EXE's filename.
But unless you want me to test it, I won't be downloading the -win package again, since the web package runs straight in the itch client without opening my default browser. (If you do want me to test, I'd be happy to.)
*So instead of ".\Housepets Christmas\game.exe", you may want ".\game.exe" or ".\12DaysofHousepets.exe".
**Or "instead" - You Have 293 Keys has a main folder in its ZIP but the itch client knows what to do with ".\293_Windows\293Keys.exe".