"Mit dem Mädchen spielt die Katze."
Katze is a feminine noun. "Die" is the article for singular feminine nouns and the article for plural nouns no matter what gender they are. Here are some examples.
-Frau take the article "die" because Frau is a feminine noun, not because it "became plural." Frauen take the article "die" because it is a plural noun.
-Apfel take the article "der" because it is a masculine noun. Äpfel take the article "die" because it is plural even though its gender is still masculine. Äpfel does not "become female" when there are more than one apple.
-Mädchen takes the article "das" when it is singular because it is neuter. When there are more than one Mädchen, it takes the article "die" because, even though the word stays the same for singular and plural, the article needs to change to reflect that there are more than one girl.
Also, the cat is the subject (nominative case) in this sentence. The girl is the indirect object (dative case). You can tell the girl is the indirect object because of the preposition in front of it, "mit," and the article for the girl, "dem."