Not normally, no. It’s officers are still commissioned Army officers, though.
Nossir, you are wrong about that. There was a big dustup at West Point that I was witness to, on just this issue.
Officers of the Army National Guard are commissioned by the various States. They are also Federally Recognized if they meet certain standards stipulated by the Department of Defense. Once mobilized and federalized, their authority and functions are exactly the same as any other commissioned officer.
Officers of the State Guard, unorganized militia, or various other names are tendered commissions by the State, but are not federally recognized, are not members of the Army of the United States, and perform no Federal function. The operate completely under State control. They have no authority to act beyond the boundaries of the State that commissioned them.