The “its” here is ambiguous.
If “its” refers to the USA, there actually isn’t a formally correct answer. Informally, Washington state evokes Washington, DC. But the US capital’s name is officially the District of Columbia. The District was originally home to multiple cities, including Alexandria, Georgetown, and Washington. Alexandria was returned to Virginia, and the District of Columbia Organic Act of 1871 repealed the charters of the cities of Washington and Georgetown and officially established a single government as the District of Columbia. The earlier name of the City of Washington is still obviously broadly used as a synecdoche for the entire District.
If the “its” refers to “state”, then there are two states that could qualify. Both Oklahoma (Oklahoma City) and Indiana (Indianapolis, from the Greek root for Indiana City) have a state name that is a subset of the state capital’s full name.