I think it's because this sentence is short for "My brother and my sister sing", so each possessive pronoun is connected to a noun in its singular form. And if you leave out the second definite article the first article will still stick to the first noun.
Ok, i got your idea. But still it's plural. Btw, how do you think it will be if words have different genders (et brød + en ost): mit brød og mit ost er på tallerken?
Yes, taken together, "brother and sister" are plural, but not the possessive pronoun before "brother". A plural pronoun expression would be: "Mine brødre og søstre synger". As you can see, the possessive pronoun takes the number of the noun it is referring to. The second "mine" is simply omitted, just like the second "min" in my previous posting. Concerning your example I would write: "Mit brød og min ost er på tallerkenen" because in "Mit brød og ost er på tallerkenen" the "Mit" does not fit to the gender of "ost". It is similar as in German. Compare "Mein Brot und Käse sind auf dem Teller" to "Mein Brot und meine Butter sind auf dem Teller". In the second sentence you need to use "meine" before "Butter" since the "Mein" does not fit to the gender of "Butter". Did you notice the "sind"? This is the plural form of "ist" (3rd person singular of the present tense of "sein"), which shows that "Brot und Käse", taken together, are plural. "Mein Brot und Käse ist auf dem Teller" would be wrong.