If the MACUSA was founded in 1693 if I remember right, probably using a different name since there wasn't any USA yet, it would have been when the British colonies in North America were colonies of the English government, and the Kingdom of England in 1693 had ministries instead of departments.
So the magical government that later used the name MACUSA could have created a ministry to interact with the famous governments of the English colonies when necessary. And if thy did so before the American Revolution in 1775-1783 they would have called it a ministry. And maybe they didn't change the name of the ministry to department after the federal government of the USA was established in 1789.
Perhaps they wanted to avoid following the latest trends in muggle government and remain detached from muggle practices.