The use of the word 'really' is one of those words that can have condescending tones if used in a certain way. I.e it implies you don't believe the person, or its a word that implies exaggeration or extension of the underlying phrase. There is probably a more accurate dictionary mechanism here that describes this better than I am.
We can all agree the question was asked poorly, but is SH racist?
But was it just a poor choice of words, in the midst of a busy room full of people who SH is having to try and converse with, perhaps not really enjoying doing so, to make small talk?
Condescending - yes absolutely seems so from the transcript.
Pushy - yes
A bit rude and/or arrogant - yes
No one can know if SH is racist unless you can read her thoughts somehow. Old fashioned use of language (for someone involved in senior monarchy (i.e toff) circles for long, is probably quite explainable, not right, but explainable) does not mean someone is racist or even stereotyping someone.