"The cat is drinking its milk."
Translation:Die Katze trinkt ihre Milch.
Yes, if I use the word Katze for it.
If I specifically wanted to point out that it's a male cat, I would use der Kater instead, which is a masculine word: Der Kater trinkt seine Milch.
But that would be like "The tomcat is drinking its milk" -- a possible sentence, but usually you don't specify the sex of the cat when it's not important, and you would just use "the cat" (die Katze) rather than specifically "the tomcat" (der Kater).
When you're referring back to a particular noun, the pronoun has to agree with the grammatical gender of the noun.
In this case, "it" refers to the cat, and since Katze is feminine, "its" has to be ihre here.
If it has been masculine (e.g. der Hund) or neuter (e.g. das Pferd), then "its" would be seine.