The wording is a bit misleading.
Downloading a file from Plex Web is indeed only available to the server admin and it’s got nothing to do with the Downloads feature. The button you see will actually download the plain file while the downloads feature is used to access media from supported Plex apps while being offline. Those “downloaded” files will show in a separate Downloads section inside Plex.
Let your friend check if they can download items to the Plex for Mac/Windows app or using Plex for iOS/Android (mobile). Keep in mind your friend needs a Plex Pass to be able to use the Downloads feature.
It does but silly me - I went to Settings in Plex and noticed “Hey I can enable downloads for users! Cool” and toggled it on. Didn’t think I need to research the topic beforehand. I had the download capability on my user and it looked like they extended that to other users.
And I think the little added “(Your friend will need a Plex Pass to download)” would eliminate the problem for other people who stumble across this. Hopefully the Plex devs will pick this up and implement it. It uber simple (just a text addition).