My understanding is the copyright infringement notices are for sharing content, not downloading. Torrent software would allow sharing by default. The media companies run nodes that download their content from torrents (where they can grab your IP), then verify the bytes they got from you in fact violate their copyright, and then send out the notice to your ISP.
If one were to stream "illegal" content, the only way media companies could find your IP is if they somehow got a warrant to raid the physical servers hosting the streaming content, and got your IP from the logs. While this is in the realm of possibility, I doubt this scenario has actually happened.
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