In fact, the UK is the same.
Although in the media, and colloquially, the head of the Ministry of Defence is often referred to as the "Minister" he is officially The Secretary of State for Defence. There is in fact a hierarchy of these appointments which is relevant to most other government departments:
The Secretary of State for Defence
Minister of State for Defence (often given a specific portfolio such as procurement)
Parliamentary Under-Secretary of State for Defence (again, with specific responsibilities like veterans etc)
It is worth noting you only have one Secretary of State for each department and multiple Ministers and Under-Secretaries. These two positions often draw from both the Commons and the Lords while the head of a department is rarely from the latter.
Historically these posts would be occupied by a senior Minister which is a much more historical title than Secretary of State which only starts to appear later.