Dover has certainly had very extensive Royal connections through the ages, indeed Dover Harbour Board saw fit to commemorate the 400th anniversary of the Royal Charter by publishing a book entitled "Dover Harbour - Royal Gateway" by Derek Leach.
Unfortunately, giving any institution the appelation "Royal" appears to be at the somewhat random whim of the monarch of the day and the only harbour to be honoured in this way is our neighbour Ramsgate.
"In 1820, King George IV embarked, with the Royal Squadron, at Ramsgate on his way to Hanover. In appreciation of the hospitality he received, he decreed that Ramsgate Harbour should have the right to add 'Royal' to its name. This status of Royal Harbour is unique in mainland Britain."
http://www.cinqueports.org/ramsgate.htm