Why is the "Configurations" option not displayed for an Azure DevOps user?
I can think of two issues that might also be the cause of your issue, but none that have to do with the DevOps Test Plan Servies directly.
1. User is a member of a group that has a deny set. (This happened to one of our users for some settings in the Wiki, as some groups had a deny set, even though the user was a collections administrator and the view for the actual permissions of the user was not giving correct effective permissions. Check your other users if they have other group memberships or are missing group memberships compared to problematic user)
2. The user does not have the license enabled. (Happened to us with the NuGet-Repository, where some users where not added with their Visual Studio Subscriber License but only basic and could therefore not access the internal NuGet-Respository. It was the only feature that required a visual studio subscriber license for us and because we still had free basic licenses when adding the user to the system, the person in charge forgot to also change the access level and only added him to the system but not with the right license -> yourorgnamehere.visualstudio.com/_settings/users -> check the user's access level, we were lucky and the user got the html error code 402 = payment required that led us to the right track of the issue)
This turned out to be caused by a missing extension in Azure DevOps. We needed to install the Test Manager extension and then assign users to the extension.
Although I still don't understand why some users were able to see the Configuration menu without holding a license for Test Manager; I assume it had something to do with holding a license for Visual Studio Enterprise while others only had licenses for Visual Studio Professional.