Now I have had a look at the HTTP requests. What happens is not the same as when I try to log in with the wrong password. In summary, the following happens:
Request: POST index.php, what is striking here is the cookie line of the request, in which the PHPSESSID variable is set twice with two different values!
Response: 302 to customer_index.php, Set-Cookie: PHPSESSID=(new value)
Request: customer_index.php, Cookie: PHPSESSID set twice again
Response: 302 to index.php
No error message appears, as after entering an incorrect password,
After deleting the cookies in the browser, the login also worked under Windows!
Conclusion: something messes up the Session Cookie.