God and man as male and female implications for gender identity

If God is both genders - or contains both genders - shouldn't people be free to choose their own gender identity? Let me explain what I mean. Focus on the Family claims that its views on gender identity are based on a Scripture verse that says, "God created man in His own image" and then goes on to assert that this "image" includes both "male and female." Correct me if I'm wrong, but doesn't this mean that God is androgynous - male and female at the same time? If this is true, wouldn't it further suggest that human beings made "in His image" have the prerogative of being either male or female - or perhaps both male and female - according to personal preference?