Lightroom ignores the Picture Style and doesn't attempt to "process" it, so I imagine it ignores it in the Exif as well. And (I presume) Windows does as well. In fact, Lightroom (as well as your generic Windows Properties view) don't show a lot of things that one could wish for if on was an Exif/Metadata junkie
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Photoshop/Bridge actually have have a more comprehensive Exif presentation. But, here's the deal: things like the Picture Style and other in-camera settings are in a special Exif section called Maker's Notes. And Adobe does very little poking around in there and I'd imagine Windows does even less.
So, if you just get Info with DPP and try to find some of the little odds and ends that you see there even in the Photoshop exif viewer, you will find that a fair number don't show up.
So the question is, does it actually matter whether Lightroom shots the Picture Style? I guess if you are shooting jpegs it could matter (in that case I'd just keep DPP at the ready). For Raw files it matters not a whit to me because LR ignores the Picture Style anyway.
One more thing, though: there are standalone Exif viewing apps that actually do delve into these "Maker's Notes" fields, with varying degress of success. So, I downloaded one of these apps not long ago, called PictureME.
And it shows a whole bunch of stuff, a lot of it repetitive, including a Whole lot from the Makers'/Manufacturer Notes.
And yes, buried deep in hundreds of entries in that section, I actually saw one for the Picture Style! I haven't experimented with it, for this pic it was set to "Standard" but I haven't cecked any recent image. In fact these days my cameras that I do most of my "serious" shooting with are set to the Neutral Picture Style and are tweaked for lower Contrast and Saturation, and I don't know whether this software goes that far, although DPP definitely does. DPP also sows your Custom Functions settings, which is nice to know!