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Hey @Twitch / @TwitchSupport bit of real/legal talk here...
Could you make it part of your TOS that making an account on the site makes the user agree that if the username is shown/spoke on a stream, that the user is OK with this and that it would fall outside privacy stuff? 
Would not work, GDPR overrules ToS and allows people to ask for data removal at any point. Certain laws require data saving for some time (for example a bank needs to keep certain data for 10years) but something like this would never fly.
Twitch might be able to refuse under one of the exemptions of GDPR around archiving and public interest. According to the ICO an exemption can be applied "When erasing your data would prejudice scientific or historical research, or archiving that is in the public interest."
Some kids 12 year old twitch name is not public interest...
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