And my family insists that it's a no because she is not royal and she has the title of Duchess of Cambridge instead of princess.
Her official style is now HRH, so she is royal, although not born royal.
British royal spouses are not created princesses, like royal spouses on the Continent are nowadays. They are thus only reffered to as princesses if they are married to the Prince of Wales (courtesy title as "princess consort") or are princesses in their own right (like the late Princess Marina). Otherwise they are referred to by a courtesy title from their husband's dukedom (like Catherine) or, if he doesn't have a dukedom, by a courtesy style based on their husband's princiary title (Princess Michael (of Kent)).
The last style was in common usage in all Europe in the 19th century, also for born princesses, and that's why you will see QV's daughter referred to as "the Princess Henry (of Battenberg") or her granddaughter Victoria Melita as "the Grand Duchess Cyril of Russia", just as any ordinary married Jane née Miss (Jane) Jones would be referred to as Mrs. Smith. or, if her husband had a high-ranking official position, (especially on the Continent) Mrs. General Smythe / Madame la générale Lefèbvre / Frau Generalin Schmiedth / мадам генеральша Kузнецовa.
