[Sundae]

Not really, market stalls have adopted the term in order to imply freshness and customer choice.
It makes them look more honest, as it means they're not selecting the mankiest ones and bundling them into a bag so you get home before you realise.

Pick your own is generally understood to be picking growing fruit at a farm. Although we probably have a shorter growing season than much of the US.
Note, the strawberries in the picture are not English. I hold out for the real thing (Waitrose do them best, and I'm not saying that because I used to work there).
Raspberries should be Scottish, ideally, strawberries English and apples French.

On a tangent, when I was 14-15 I worked at a pizza restaurant and one of the Managers was called Punit, he was a real stickler for detail. We called him Strawberry behind his back, because we were still schoolgirls and didn't really want a profession in the hospitality sector.