He is the “Commander in Chief”.
That’s one of the constitutional functions of his office. It is not his job title. He is the President of the United States. And it is also worth noting that when he receives military salutes, including on state occasions a 21 gun salute, it is not the CnC that they are saluting, it is the head of state. The founding fathers were absolutely adamant that the presidency was to be a civilian office.