Can the Navy/Army/Air Force/etc call your phone and show up as another contact?

Usually it will show up on caller ID as US Government.
If the recruiter is using their issue cell phone, that phone may come up as another user. They are issued by recruiting headquarters to individual recruiters. Example: My recruiting HQ was in North Las Vegas, NV - I was in California. Usually they will have toll free numbers.

If the recruiter is using a personal cell, they may have gotten that from another state on a previous assignment.

So if he called once from the office, once from the Government cell, and once from his personal cell, you'd see three different things.

However, if your friend was dodging calls, the recruiter may have borrowed other cell phones also. They're a sneaky bunch!! lol...