Fixed!
How I found it:
I downloaded the newest IPB version, started Sublime Text 3 and searched for 3S290/3, found one file which was located in "root/system/Dispatcher/Api.php".
On line 187 you see it shifts the array of URL-bits, then it checks for a regex and throws the error. I commented the exception throw and used print_r/var_dump to check what the variable $app contains. Surprisingly the array was empty.
I then checked what $pathBits does and it seemed as expected, that it doesn't apply any of the URL-bits to the variable. (reason for that remains unknown to me, I'm not a IPB dev nor a plugin-freak. So I just did a workaround to fix my issue.)
Solution:
A small workaround on line 84 fixed that issue.
/* Get the requested URI */
$requestURI = $_SERVER['REQUEST_URI'];
/* Remove /api/ from the bits */
$requestURI = str_replace("/api/", "", $requestURI);
/* Finally apply the new URL-bits to $pathBits */
$pathBits = array_filter( explode( '/', $requestURI ) );