Alright, I tested Lily's Day Off with Windows.
It is listed in my Itch app library but it still asks me to buy it so there must be a bug between your visual novel and the Itch app as the VN is listed in the library so the Itch app does acknowledge that I own it but it doesn't want to install it so I suggest to check how your VN gets along with the Itch app.
I was able to download and read your VN from Itch website without using the Itch app so the issue definitely is related to the Itch app.
But even though, I was not able to launch your VN from Linux Mint no matter with or without the Itch app so I would also say that your have to check for yourself Linux compatibility before selling something for Linux...