“Washington, DC” means “Washington, which is located in the District of Columbia”. The fact that the two are the exact same area was not always so.
Before it was completely urbanized, there were unincorporated towns within it. Only the Federal gov’t area was “Washington”, and it wasn’t incorporated. That is, it didn’t have a mayor, city council, municipal entities. The US Congress ruled it and DC totally.
Georgetown, MD, was a thriving port before DC was created around it. So to was Alexandria, VA. Other areas of DC still carry neighborhood names that were derived from the towns that used to be there.
[Originally, DC was a 10 mile square (100 sq miles) diamond with 2/3 on the east side of the Potomac and 1/3 on the west. Because forming a local gov’t for it was too much of a pain for an area that didn’t have many residents, the eastern part was subject to Maryland laws and the western part to Virginia laws. Then in 1847 (IIRC) the western part was ceded back to Virginia. By that time, Fairfax County had adjusted to not having its eastern part. So the ceded land became Alexandria County and City of Alexandria. (Cities and counties in VA are separate entities, even if the latter surrounds the former.) Later still, in honor of Lee’s Arlington House that overlooked the Potomac and Washington, Alexandria County became Arlington County.]
The remaining part of DC became urbanized. (So much so that many street names in DC extend for miles into Maryland.) With this, the definition of Washington city grew too. Finally, what was DC was also Washington.