Sounds like permissions issue. Trying it on firefox or chrome (which ever is the opposite of what you are using) will very likely fix it.
However I have been looking into a new method of getting the players name. So in the next version this issue should be fully fixed.
Are you running Linux? On Linux (and possibly on other systems; I'm not able to test such systems myself), certain components of web games (most notably the "prompt()" function) don't function correctly when run from the itch.io client. It seems to be due to an, I would argue, poor engineering decision by the team who made the backend Itch uses; see https://github.com/electron/electron/issues/472 for more info.
...Frustratingly, the "prompt()" function just so happens to be a /very/ common tool among HTML game makers to prompt the user for a custom name, so... this lack of support breaks quite a few furry games. I use the same workaround you found; if the workaround bugs you too much, see if the game functions as intended in a standard web browser.
I have seen another way to imput names. So I think I will attempt to migrate over to that method. So no more prompt at all. I will try to get it implemented for the next update.
And thankyou for pointing this out. I had been considering this change for a while and your comment has cemented needing this change.
With any luck it wont ALSO break in linux.