Not necessarily. Capital is a more broad term, just referring to the capital of any place. It could be a country, state, province, organization, etc. State capital is more specific, referring to the capital of a state.
The state of Oaxaca (both state and its capital have the same name).
Oklahoma has about the same capital as the state name. Its capital is Oklahoma City.
The city of Aguascalientes is the capital of the Mexican state of Aguascalientes.
Austin, Boston
There is no strait and capital with the same name.
The Indian state capital that starts and ends with the same alphabet is Ahmedabad, the capital of Gujarat. It begins and ends with the letter 'A'.
The city of Oaxaca (both the state and its capital have the same name).
washington
Guanajuato also. Guanajuato is a state and a city (in the same state)
Luxembourg has no real meaning. It is the name of a city (and capital) of the small state of the same name, nudged between Belgium, France and Germany.
.All the state offices are there (or the headquarters are there), the legislation is handled there, etc. It's the same reason Washington DC is important as the capital of the US. A state capital has the same role, except on a smaller scale. It is the State Capital... Enough said.
Columbus