Not technically. Yes, they’re the Commander in Chief, but they maintain civilian status. They oversee the military, but are not actually a member of the military. This is also why they cannot be prosecuted under military law or court martialed.

Can’t find current link on Defense website, but here’s an older one: https://web.archive.org/web/20170512070110/https://archive.defense.gov/news/newsarticle.aspx?id=45870