[jkaplowitz]

True in most contexts, but not in the context of state-level legislative language where it would usually refer to that state’s official role of that name. Most equivalent US legislation to what we’re discussing here would occur at the state level, since incorporation in the US is generally handled by the states. (The US federal government does track companies in various ways, of course, but the publicly accessible company registers come from the states.)

[omnicognate]

The context here is a UK law, not US state-level legislation, so I don't see the relevance. And the similarity between the UK and state-level US meanings of "Secretary of State" was overstated anyway. There is no one Secretary of State in the UK and it isn't a specific position in its own right. There are 17 Secretaries of State, all covering different things. The legislation here refers (I think) to the Secretary of State for Business and Commerce rather than, for example, the Secretary of State for Culture, Media and Sport or the Secretary of State for Education.