Princess is definitely more formal than say Carnival or NCL. I just returned from the Carnival Glory and was surprised at the way people dressed. In spite of reading the boards and comments, I was still somewhat amazed. I was on the Brilliance in March and did not see that many real casual people at dinner but it was a 10 day cruise.
I've been on 6 cruises in 3 yrs., 5 different lines, and I realize times are changing, but people on Carnival pushed the button to the max. I'm not talking young folks, or any particular group, just overall appearance was very casual. We sat with an older woman who wore shorts to dinner several nights, didn't change my dinner any but it did surprise me.