Translation:The child is playing with his lunch.
It's extremely common for German speakers. I once worked at a camp whose director, from NYC, had "gone native" in the Austrian countryside as an adult and married a German. This usage had even crept into his English: he constantly referred to a child forgetting its towel at the beach or wishing it could go home. However, he never used this when referring to a specific child whose gender we knew.
"Their" is plural while "the child" is singular, so that is not an appropriate match. "Das Kind" is neuter gender and so should be referred to with "it" if the child's gender is unknown. We say "The child is playing with their lunch" a lot in English because 1) we don't like to refer to a child as "it." 2) while referring to a singular, unknown child in English, "he" is grammatically correct (last thing I knew). However, political correctness frowns upon this, but saying s/he is burdensome. Thus, people started referring to a singular, unknown person with "their" to avoid the gender issue.
This sentence could also be translated as "The child is playing with his lunch."
In this case, "seinem" means its. In German, when using a pronoun or possessive pronoun in place of a common noun, the noun's gender determines whether you refer to it as "er" (masculine), "sie" (feminine), or "es" (neuter). "Mädchen" is neuter, so you would refer to it as "es", and the possessive pronoun of "es" is "sein".
I do not know why Mädchen is neuter, but I think it has to do with the structure of the word as opposed to its meaning. To answer your second question, "sie" is NOT the subject of this sentence; the subject of this sentence is "das Mädchen" or "es". In English, we determine which pronoun to use (he, she, or it), based on the definition of the word (natural gender), whereas in German, they determine which pronoun to use based on the gender of the word (grammatical gender). Because "Mädchen" is neuter, the pronoun you would use to refer to it is "es", NOT "sie".