It is actually important to have the possessive pronoun in front of each word to emphasize that the roles are applying to two different persons. Usually that would be clear from the number of the verb, but in casual language the following is conceivable, say, at a garden party: "Und wer ist das da hinten?" "Das ist meine Tochter und meine Frau." The answer "Das ist meine Tochter und Frau" might get you funny looks.
A more benign example may be to introduce somebody with "dies ist mein Freund und Nachbar". Here omitting the second pronoun for a single person seems natural. Generally omitting the second pronoun conveys to me the sense that the two roles are closely connected, perhaps from the Biblical passages quoted elsewhere in this thread.
None of this is, however, clear-cut, and will usually be obvious from context.
In your specific example a grammatical factor comes into play: If case, number or — for singular — gender differ between the two nouns it is actually mandatory to repeat the possessive pronoun, like it would be with an adjective: Das sind meine Frau und mein Kind. You can combine male and female plurals though, as in "das sind meine Söhne und Töchter".