Even if the setting is for
anyone, that doesn't mean that they don't need to sign in.
The only way you can avoid the requirement of someone signing into a Web App with a Google Account,
is to publish it to run as you (ME) and anyone, even anonymous
Otherwise, they must be signed in with a Google account.
If they aren't signed in, then they must sign in.
There is an issue when someone is logged into multiple Google accounts in the browser.
Apps Script can encounter an error in that situation, because it can't determine which account to use.
Who knows if that will ever get fixed.
Google has totally failed on that issue.
But, that doesn't seem to be your problem.