It's not, by its own admission, a Government in its own right, but just a ministry:
(in certain countries) a government department headed by a minister.
"the Ministry of Defence" synonyms: government department, department, bureau, agency, office
"the ministry for foreign affairs"
Did JKR intend to make the MoM part of the British Government? Or is this a mistake of language?
I do recognise that the Muggle PM in the later books does not recognise Scrimgeour, but I don't find this conclusive (Wizards could be classified above the PM, or just be on a Need to Know basis)