There are ways to lead that have nothing to do with your title, and if you know how to use those skills, you'll be much more productive. Most of follower positions still require you to interface with with other people and leadership skills make this so much easier.
Are you preparing project plans proactively? Are you issuing regular status updates to stakeholders? Are you offering to help people where you know there's benefit to be gained to the company? Are you asking the right questions at the right time to the right people? You don't need an office and a title to do these things, everyone can do them, and the best employees do.
Now you've also been specifically asked about things like "supervising" and "subordinates" - these are not leadership qualities. These are managerial tasks. Don't confuse the two! A leader can help resolve conflicts between any two people, but only a manager should be "resolving conflicts between subordinates", and the manager has the authority to order a resolution (which one might argue is a piss-poor way to resolve a conflict). A leader on the other hand can't order anyone to do anything, instead relying on persuasion to resolve the conflict. (That being said, a great manager will use leadership skills instead of managerial authority whenever possible!)
Now why would they ask these types of questions? Well, the first type has a couple reasons. Most obvious, they want to know you'll be capable of taking care of yourself and that you'll be able to take down roadblocks when necessary. Also generally, people with those skills take control of their own destiny, and they take extra care to be prepared and in general are just more productive employees.
The specific managerial questions are a bit odd, but it's nice to know if you have them. Experience in a managerial role gives you a different perspective on being an employee (one that is usually a net positive for the company).
Bottom line, you don't have to be a manager to be a leader, and experience in either manager or leader roles are likely to make you more productive at any level of employment.