The federal capital actually is named Washington, the state equivalent is the District of Columbia. Denver is to CO as Washington is to DC. It just so happens that the City of Washington is coterminous with the District of Columbia, so that is where the confusion arises.
That would make sense, but it's not accurate. There once was a City of Washington established by Congress in a portion of the District of Columbia. The City of Washington was disincorporated by Congress in 1871, leaving only one remaining official entity, the District of Columbia.
https://en.wikipedia.org/wiki/District_of_Columbia_Organic_Act_of_1871