Is it correct to say 'me and she are family'?

No, it is not correct.

The first person pronoun 'me' is an object pronoun used for the subject of the sentence. The first person subject pronoun is 'I'. It is also customary to put the first person pronoun last in a compound subject or object. The correct sentence is:

"She and I are family."

An example of a compound object of a sentence:

"The family invited her and me.