This is wrong: try downloading something with Safari, which blocks third party cookies by default, and it works fine. Instead, the problem is that downloading from Google Drive is using User-Agent sniffing to determine whether third party cookies are expected to be enabled, and choosing between implementations. (Disclosure: I used to work at Google, but I don't know anything internal on this)
Oh wow - so while there was a non-malicious explanation in another comment ( https://news.ycombinator.com/item?id=32149130 ), this clearly looks like malicious behavior. They're relying on the fact that most browsers lack per-domain cookie controls to force Google Drive users to allow third-party cookies knowing full well the majority won't remember (or bother) to disable them after.
I'm not sure why they don't use the new flow for everyone. My guess is that it's less secure? Maybe that if the link they generate is shared it gives access beyond what the original owner chose to share?