We have not hit a port that requires cruisers to carry their passports. We did surrender them to the ship when going between Greece and Turkey and back for inspection by border patrol of those countries. I understand there is a similar process when going to St. Peterburgh though we have not been.
Some ports will require you to carry ID in addition to your sign & sail card. Actual enforcement may vary. Pay a attention to announcements and the Fun Times and check posted signage as you disembark.
Our passports always cruise in the safe.
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
Star Clippers: Royal Clipper 2015