"Commander", yes, but they command very little. During World War II, President Roosevelt gave exactly two direct commands to the military - that they had to divert 50 long-range bombers to help protect Atlantic convoys, and that North Africa be invaded in 1942. In a four-year-long war, he gave two commands. Contrast that with Hitler, who micromanaged his military, and did them no good by doing so.

So, yes, he does have final command (subject to "legal orders" and similar restrictions). But he's not commanding in the normal sense of a military commander.