Good question. I don't know when Asian women started to prize the double lid so perhaps it was only after Europeans arrived on their shores? Yet it's also possible that it's not related to Europeans at all. There are Asians who naturally have the double lid, but plenty of us do not.
Many cultures enjoy setting beauty standards to differentiate the "elite" from other people. Colorism has long been an issue in Asian cultures because lighter skin implies you have the wealth to not toil outside in the sun. It's only in recent years (e.g., Japan) where being tanned is now seen as a sign of wealth because you're not toiling inside an office all the time.