How to achieve, that a voter can only vote once on a quiz?

Hello Clemens and thank you for posting.
The playing limit does exactly what you guessed. It defines how many times can each user or visitor take a quiz.
Everything is simple when it comes to logged-in users. They have a unique ID in the database, so limiting play is an easy task.
The tricky part comes when we're trying to limit a visitor from taking a quiz multiple times. Since this is a visitor, we have no back-end information that could help us identify him.

SimpleQuiz uses a combination of methods to TRY to limit quiz play: Session and cookies.
SimpleQuiz uses the session to assign a unique ID to each visitor, and leaves a cookie after quiz play.
When the quiz checks if this visitor has taken the quiz before, it prefers the database result datetime over the cookie, as per the discussion in this forum post .
Of course, someone can clear their cookies, and play again. They could obviously also write fake emails. If you have any further suggestion on applying this limit, feel free to say it.

SimpleQuiz doesn't log the user IP and doesn't compare the name/email pair in order to apply this playing limit.
Please make sure that you don't have any template overrides that may be blocking the latest updates.
I just tried it online at demo.mavrosxristoforos.com/administrator and set the Simple Survey's playing limit to Only once. It works correctly.
So, if it doesn't work for you, we may need to do some debugging.

Thanks again.

Unfortunately, the check for whether a quiz can be re-taken or not, is currently done before starting the quiz. Therefore, the user hasn't completed any information.
Adding this extra layer would complicate the finishing process a little bit.
If you want, we can examine it and see if it's possible.
Please send me an email, as per your other two posts, in order to see what's wrong.
I believe that all these concurrent issues may have a very simple explanation.
Thanks again.

After checking, there seems to be a problem with the user session in both the front-end and the back-end.
While browsing the front-end of your site, the session cookie gets purged. Then, a couple of requests seem to go with no session cookie at all, and then, a couple of requests later, a new session cookie appears.
You can observe this behavior, by using Chrome Dev Tools (F12) and going to the Application tab.
In Storage, select Cookies and then your site's URL.
You will see that a session cookie appears, then disappears, then re-appears with a different value.
This causes SimpleQuiz (and Joomla in general) to believe that this is a new user.