Is tennis perceived as a posh sport in the US?
In the UK, while there's nothing stopping a person from playing whenever they like at their local leisure centre, the main tennis social calendar in England is more of an "upper class person has a day out" thing. It's socially aspirational, if you're that way inclined.
(In contrast, something like F1 is very much a rich man's sport - but not necessarily posh.)
So I tend to see Anna's obsession with tennis as coming from the same place as fashion's love affair with aristo models like Stella Tennant, Cara Delevingne, Jean Campbell etc.
But in repackaging it for an American market, what does tennis mean to that audience? Achievement, certainly, but does it come with other significance? Is there still the sense that an interest in tennis might not come from a love of sport - but from what the sport represents?