Why did Washington name Washington state Washington?

He didn't. The Monticello Convention of 1852 chose that name.

They wanted to have at least one state named after a president. George Washington had been dead for over 50 years before the state was named.

Oddly, the first choice of names was "Columbia", but it was felt that it would be confused with the "District of Columbia", so it was named Washington instead, in 1852. It did not become an official state until 1889.