This sounds rather like song lyrics to me.
Personally, if you don't pursue formal grammar, the two sentences would still function. The difference lies not in the meaning. The first sentence uses "做", a verb, an action, while the second one uses "是", in Chinese we call it BE verb, a declaration of a status of being.
”我做我的王“ is more like speaking of an action, meaning "I am going to be my own king/I am going to follow my heart". “我是我的王” is more of a claim of the self. And you can say ”你做我的王“, which means "please be my king".
Hope this makes sense.