See #3171458: Media Library and Field groups not compatible → and the latest patches. The fixes also have to be implemented for oembed form as this seems to be an exclusive alternative to upload form.
See https://www.drupal.org/node/3089217 →
As that issue is already closed, I'm opening a new one here for the additions.
Needs review
3.0
Code
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It's likely this issue predates Contrib.social: some issue and comment data are missing.
We should probably use hook_form_BASE_FORM_ID_alter() to target all media library add forms.
Applied the patch, and now I have working field groups in my media library! Sweet!
Don't know if it is too generic now, but it works and clicking around the site a bit, I don't see it breaking anything.