Archie cannot become a prince due to the 1917 decree by King George V, which only gives HRH titles to those in the direct line of a future monarch.
I News reported that according to King George V’s 1917 patent, only royal offspring in the direct line of succession can receive the title of prince or princess. Only the sons and daughters of the monarch and children of sons of monarchs can claim these titles. Eugenie and Beatrice are princesses because they are the daughters of Prince Andrew. As Prince Charles's sons, Harry and William are princes.
Others have noted that Archie should be a prince, as the Duke and Duchess of Cambridge's children have HRH titles. However, they are great-grandchildren of the Queen and will one day will be grandchildren of a future monarch, whereas Archie is not. Princess Eugenie's baby son is not eligible for an HRH title for the same reasons. Archie will be able to gain the title of a prince once the Prince of Wales accedes the throne.
In 2013, Queen Elizabeth II amended the order that any of Prince William's children could receive an HRH title.
On March 4, 2021, Prince Harry and Meghan Markle were interviewed by Oprah Winfrey. During the interview, Harry and Meghan shared their experiences of being part of the Royal Family. The Duchess of Sussex said that she felt she had been treated differently and that her son would be unable to receive the same privileges because of his race.
The Associated Press reported that Buckingham Palace released a statement after the interview that Her Majesty the Queen grieved the situation of Megan and Harry. They added, "The issues raised, particularly that of race, are concerning. While some recollections may vary, they are taken very seriously and will be addressed by the family privately."
It is not possible to check these claims, due to the tightly guarded nature of the Royal Family. It can be noted that many commentators have expressed sympathy towards to the Duchess of Sussex and found that she was treated differently to the Duchess of Cambridge, Kate Middleton, in the UK press. A comparison of headlines in Buzzfeed suggests that Kate Middleton may have been treated in a more favorable light.
However, as it stand, Archie cannot be entitled as Prince due to the 1917 decree by King George V. In order for Archie to get the title, Prince Charles should become the King or Prince of Wales accedes to the throne. This specific claim is a formality and there is no evidence to claim that it is a result of racism.