Actually, the term "secretary" is used for offices in the United Kingdom as well: Secretary of State, Home Secretary, Chief Secretary, and so on.

The word comes from the Latin secretarius, literally meaning a person entrusted with a secret, or a confidential officer. The term was first used as a title for ministers presiding over executive departments during the reign of Elizabeth I - e.g., Chief Secretary for Ireland.

There are other European countries that still use the term Secretary of State.

From the Oxford English Dictionary:

The occurrence of the title ‘(Principal) Secretary of State (†Estate)’ under Queen Elizabeth may be taken as indicating the beginning of the development by which the king's secretary... became a minister invested with governing functions. Throughout the 17th c. there were two officials jointly holding the office of Secretary of State, and in the 18th c. the number varied between two and three; till near the close of this period the two (or two of the three) were distinguished as ‘Principal Secretary of State for the Southern Province’ and ‘Principal Secretary of State for the Northern Province’, with reference to the division between them of the control of foreign relations... ; but with regard to internal administration no division of functions was formally recognized. At the end of the 18th c. there were three Secretaries of State, and shortly afterwards the division of functions between them was recognized in their official designation, as ‘Secretary of State for Home Affairs’, ‘for Foreign Affairs’, and ‘for the Colonies’. In 1854 a Secretary of State for War was added, and 1858 a Secretary of State for India. The Secretaries of State are often more briefly called the Home Secretary, the Foreign Secretary, etc. The Chief Secretary for Ireland (officially styled the Secretary to the Lord Lieutenant, and informally the Irish Secretary), and the Secretary for Scotland (first appointed in 1885) were not secretaries of state, but had similar functions, and were (c 1911) members of the cabinet. The Secretary at War (down to 1855, when the office was united with that of the Secretary of State for War) was the parliamentary representative of the army, and had some degree of control over its finance. There have been numerous changes (too complex to set down here) in the nomenclature and duties of Secretaries of State since the nineteenth century. Since 1945, principal Secretaries of State have included the Secretary of State for the Home Department (Home Secretary), Foreign and Commonwealth Affairs (Foreign Secretary), Industry, Defence, Employment, and Northern Ireland. The title Chief Secretary (to the Treasury) was introduced in 1961: it is a ministerial appointment as opposed to the various civil servant Treasury Secretaries. The principal Secretary of State (usu. the Home Secretary ) is sometimes referred to as the ‘First Secretary’.

In the U.S., the Secretary of State corresponds approximately to the British Foreign Secretary. Other cabinet ministers, heads of executive departments, are the Secretary of the Treasury, of War, of the Navy, of the Interior, of Agriculture. Each state of the Union has also its Secretary of State (or a corresponding officer with some other title). In recent years, the nomenclature of senior U.S. cabinet ministers has (as with their counterparts elsewhere) been subject to extensive changes.