No - Princess and Carnival are separated companies under the same corporate umbrella. According to one source I read (one of those travel books) this is due to anti-trust concerns when Carnival bought Princess.
All I ask is a tall ship and a star to steer her by; John Masefield
Carnival: Glory 2004, Destiny 2008, Splendor 2009, Freedom 2011, Valor 2012, Dream 2013
Celebrity: Summit 2011
Princess: Ruby 2010, 2014, Caribbean 2013, Coral 2014, Regal 2014, 2016
Sail Windjammer: Mandalay 2017
Star Clippers: Royal Clipper 2015