The US Space Development Agency, focused on missile defense, is working with SpaceX on tracking and communications constellations. But the communications is using laser links for in-space communications. Military Starlink is possible but not proposed yet.
Walter Isaacson, the book's author, gave a long-form interview recently on The Lex Fridman podcast. In case you'd like to form your own opinion about the book's legitimacy, its writing/Musk-supported research is substantially discussed:
>A senior Ukrainian official has accused Elon Musk of “committing evil” after a new biography revealed details about how the business magnate ordered his Starlink satellite communications network to be turned off near the Crimean coast last year to hobble a Ukrainian drone attack on Russian warships.