I can see your point about this rchat. Beatrice and Eugenie are princesses as well as being Sarah's daughters and it wasn't necessary to make her host feel stupid by not having addressed them properly. She could've easily said, "My daughters, The Princesses Beatrice and Eugenie are doing well, thank you." Or simply "My daughter are doing well, thank you."
When you hear celebrities being interviewed the more humble ones don't insistently add their equally famous boyfriends/husbands, girlfriends/wives, parents or siblings into their interviews. When someone asks Kate Hudson how her mom is, she doesn't say "My mother, Goldie Hawn, is doing well." She just says, "My mom is doing well." Likewise for Catherine Zeta-Jones of her husband Michael Douglas. She simply says "Michael is doing well."
I think that such emphasis occurs (for royals and for celebrities alike) when there is an insecurity in the person that people might not realize who they are related to. In the business world you can see it at meeting when someone repeatedly mentions that they are the CEO or Director of X at a meeting when clearly everyone in the room would know who such individuals are.
I think at heart Sarah is at least a bit insecure and with her daughters growing up and taking to the public stage more on their own, she won't be as prominent in the media as she used to because the "field" of royals is expanding. William and Harry have always been in the public spotlight, but now Beatrice and Eugenie are venturing into the public spotlight into their own, Zara and Peter Phillips are attracting some attention with respect to their love lives, as is Gabriella Windsor, and Frederick attracts attention for his own antics. And Camilla has certainly gotten a lot more media coverage since marrying Charles and now that she is Camilla the Good, and out and about performing royal duties, all these individuals are crowding Sarah off the stage and making her less of a presence in the world. Such a loss of attention could affect her Weight Watchers contract, too, since less media attention on her means that she is less effective as a "celebrity spokesperson."