Hi @smorup, thanks for posting today!
Most likely, that user is on a Dropbox Business team on which the team admin has not permitted their team members from having edit access to external shared folders.
This is usually set up as a security precaution. That user could try to get the admin to change this setting, but it is up to their team admin, and there isn't a way to work around this, aside from inviting another Dropbox account they own to your shared folder.
If you need any more info, let me know!