I have always been very intrigued with the decades of connection and friendship of the Mountbattens and the BRF . In addition to the blood connection as descendants of Queen Victoria, it was only intensified when Elizabeth married Philip. Both Charles and Anne were close to Lord Mountbatten and his wife Edwina and stayed with them when Philip and Elizabeth would be in Malta during the Navy years.
Charles was very close to Lord Mountbatten. Like everything else, the younger royals ig William and Harry not so much. I think the ties will go away once Charles is gone and other Mountbattens of his generation. William,Harry (and the other young royals) don't seem intent on ties with their noble relatives or continental royals which is a shame. These new Boleyns hold nothing on these old school bloodlines.
Lord Mountbatten wanted the Queen to call her House and all descendants Mountbatten. The Queen Mother found him pushy and too "dynastic." The fact that the Queen and her immediate Royal heirs (ie. Charles etc) are l Windsor but others who needed a surname became Mountbatten Windsor is a testament to compromise with him. It also assuaged Prince Philip's feelings as being no more important than a "bloody amoeba."Interestingly when Princess Anne signed the marriage register for her first marriage in 1973, she signed it Mountbatten Windsor.
I've always found the Hessian/Battenberg family connections fascinating as well, starting with Queen Victoria's daughter Alice. There were certainly some interesting characters in that family! An excellent book I can highly recommend on the subject is David Duff's 'Hessian Tapestry'. I've read it a few times, never fails to entertain me.