Although the current monarch, House of Windsor, is not directly related to the House of Stuart, keep in mind that HM Queen Henrietta Maria of England; wife of HM King Charles I of England, mother of HM King Charles II of England and of James II of England, grandmother of HM Queen Mary II, and Queen Anne, was born SAR Princesse Marie Henriette de France, daughter of King Henri IV, first Bourbon King of France, Sister of King Louis XIII, and Aunt of King Louis XIV the Sun King. She is the one who introduced the custom to title the first daughter of the monarch, Princess Royal, because at the court of france the first daughter of the monarch was title Madame Royale.
In fact her first daughter HRH Princess Mary, was the first Princess Royal. HRH Princess Mary's son married Queen Mary II and went on to reign as William III of England. Her younger daughter HRH Princess Henrietta of England, married HRH Prince Philippe of France, Duke of Orleans, bother of King Louis XIV and founder of the House of Orleans, a cadet branch of the House of Bourbon. After the death of Charles I, Queen Henrietta Maria and her daughter Henrietta, lived at Palais Royal, the Official Residence of the Duke of Orleans, next to the Louvre.
Though i am stretching it a bit, the link to France is there if one want to look for it. But this boy is mainly named Louis like his father after Earl Louis Mountbatten, who was assassinated by the IRA and was very close to Prince Charles.