Every company is different. Royal Caribbean keeps a tight rein on its Celebrity and Azamara brands.
But Arison and Carnival Corp. is different. They are actually quite well known for being extremely "hands-off" when it comes to their individual cruise lines.
Of course they dictate certain things, such as when they can buy fuel in bulk, etc. But otherwise they largely allow the cruise lines to make their own decisions.
If there is any one line where Arison is most closely affiliated it is Carnival, however, because that was his original company, the one he used to finance buying the rest.