The Upper East Side was never called the "Silk Stocking District." That nickname belonged to a Lower East Side neighborhood in the late 19th century.
Wealthy New Yorkers moved into fashionable townhouses there, and the area became known for its stylish female residents who wore luxurious silk hosiery—hence "Silk Stocking District."
The Upper East Side developed later as an elite residential enclave (especially along Fifth Avenue and Park Avenue), but it carried its own distinct reputation—associated with "old money," museums, and luxury—not the Silk Stocking moniker.
So, if you hear "Silk Stocking District" and "Upper East Side" linked, it's a historical mix-up.