|Postmark from Court Post Office at Buckingham Palace (used 2019 - present)|
There are a number of different postmarks for the British Royals. The postmarks usually relate to which post office your letter was processed though.
The Royal Mail (UK's national postal service) runs the Court Post Office at various royal residences (Buckingham Palace, Windsor Castle, Sandringham, etc.). They receive all the incoming mail for the Royal Offices & Households in that area. They run additional security checks and then sort the mail and deliver it to the correct Royal office.
It is then the Royal Mail's job to collect all of the Royal's outgoing mail and take it back to the Court Post Office. It is there the outgoing mail is franked (stamped with the postage and postmark). The mail will then be sorted and sent out to be delivered.
This is why all of the royal mail from London has Buckingham Palace on the postmark; it is because that is the name of the post office where it is processed.
Since 2001, the Sovereign Grant has paid for the Royal's postage. Previously, the service was provided for free by the Royal Mail.
In March 2019, The Royals changed their PPI number. This forced them to redesign all of their postmarks.