Relena's Queen of the World dress in Mobile Suit Gundam Wing is pretty much a direct copy of one of Ann's dresses.
Episode 156 of Sailor Moon features the fictional runaway Princess Rubina, who has the same hairstyle (albeit blonde) and casual outfit as Princess Ann's.
Spring 2018 anime Tada Never Falls in Love directly references it with a copy of the Vespa scene in the opening, and in the plot: a European Princess escaping her role to experience freedom. (In this case, incognito as an exchange student in Japan.)
Gunslinger Girl. Henrietta is an enthusiastic fan of Roman Holiday, delightfully pointing out film locations when she visits Rome. Having a Precocious Crush on her handler Jose, she gets him to buy her a gelato which she eats on the Spanish Steps with him like Princess Ann did in the movie.
The band Deep Blue Something has said that this was the Audrey Hepburn film they had in mind when they wrote "Breakfast at Tiffany's" but they couldn't make it fit the meter.
from the movie.