Thanks for the feedback. This is almost always a browser issue. If you have noscript or another javascript blocker, allow itch.io and the page itself or try another browser. https://twinery.org/questions/46598/on-itch-io-error-no-valid-storage-adapters-f...
Actually, if you have a JavaScript blocker running, then it won't even get to the point where it tries to access sessionStorage, which's what is causing the error. So no, that can't be the problem.
As the link you included says, it's likely due to the browser's security settings. If you're trying to play it in a private window, then try paying it in a normal window, which has less strict security settings. If that doesn't help, then it's likely something in your browser's security settings which needs to be changed, probably under the "cookies" settings (even though sessionStorage and cookies aren't quite the same thing). Alternately, try using a different browser.