If you use an image from a free-download website and the same image is used in other websites, is this also copyright or not?
No. The copyright holder on an image is the person who originally publishes it not anyone else.
If John Doe makes a photo of Tokyo and licenses it under the Creative Commons, then it is available to everyone. Otherwise royalty free sites wouldn't exist because only the first person to use the image would be within their rights.
Another way of saying it is:
John Doe takes a photo of Tokyo and makes it available under Creative Commons. Adam Brown uses it on his website. Charles Dickens uses it on his website, AFTER (and being aware) of it being used by Adam Brown. Adam Brown has zero ownership of said image though and therefore cannot claim any rights to the image in order to prevent Charles Dickens from also using it.