Discord's developer TOS says that API users must "comply with all applicable privacy laws and regulations" and adhere to a privacy policy. It doesn't otherwise put any specific restrictions on what that policy can look like. It certainly doesn't say you can only use user content if the user "individually and permanently grants a license".
If you think that Linen would be violating the user's rights, as opposed to Discord's, then that's a separate issue, and one that would probably come to a question of fair use and/or implied license.
Out of curiosity, do you also think public mailing list archives are (or should be) forbidden, by the same reasoning?
The user doesn't implicitly grant the server owner a license to their content when they post to Discord, so reuse of that content by a third-party service has no implied consent permitting it.
I think that Linen is facing considerable legal exposure risk and is taking the Uber approach of "break the law and pray that no one sets up a class action lawsuit against them for their rights violations". Discord has set up their chat service to be private, and extracting that data and making it public without each user's consent is a violating of privacy that — unless an overriding public good is served — would lead to their fair use claim being overturned.
I expect Discourse to ban their official integration for being inappropriate for the platform (their terms of service permit them freedom of judgment in that regard), and then to warn and eventually terminate _entire Discords_ when their operators use the predictable metastasized proxy Docker image to circumvent that centralized integration ban.
Email, as opposed to Discourse, would inherit the concept of "the recipient of a letter is permitted to do whatever they want with it", same as we have had with postal mail for a very long time — as email can be reasonably considered postal mail.
(Disclaimers: I am not your lawyer, this is not legal advice. I strongly advise consulting with a lawyer on the risks and uncertainty surrounding Linen. I do not consider there to be sufficient case law to be certain what the outcome of judicial proceedings for any of the above assertions would be. My theoretical opinions and predictions may or may not be upheld in a court of law.)
If you think that Linen would be violating the user's rights, as opposed to Discord's, then that's a separate issue, and one that would probably come to a question of fair use and/or implied license.
Out of curiosity, do you also think public mailing list archives are (or should be) forbidden, by the same reasoning?