When I think of 'Western culture', I think of the culture that developed in Protestant Europe and English-speaking America from the early modern period onwards, and aspects of which are diffusing through the rest of the world. I agree that it's a bit too broad of a characterisation to really be of use, but it is a thing.
For many, many centuries. It's not anything from the last few years, nor is it "we against them".
The West is essentially all the world's nations whose cultures are derived from or greatly influenced by Latin speakers and Judeo-Christian values, which traces its influence back to Ancient Rome.
It's a very valuable and important thing to recognize, because things that people in the US and Europe often implicitly assume are human universals, aren't. They're only universal across the West, and not found in e.g. China or India.
So it's not creating division. It's recognizing difference that already exists, but that not everyone is always already aware of, because we tend to exist in our own little bubbles.
When did we become Westerners and started to think in the same way?
I recognised a trend for such expressions in the news, too, in the last few years. It's misleading and creates a "we against them" mentality.