I believe su more correctly means her (not hers) and your (not yours), while hers and yours (singular formal or plural) would be suyo, suyos, suya or suyas. I realize my knowledge of English grammar is probably not as good as yours. I know my knowledge of all things Spanish is not even close.
I have a cat who is quick to drink my milk from my cereal bowl if I get distracted.
The singular "their" as a gender neutral pronoun is an odd thing in English. It is widely used in the vernacular (because english lacks a singular gender neutral pronoun), but many "experts" in English don't believe it as proper. I myself use it, my family uses it, but I get mixed opinions from the various English professors I have had. For further reading: http://en.wikipedia.org/wiki/Singular_they#Acceptability
As Iago pointed out above, "su" can mean his, her, its, your (formal singular), their, or your (plural). It is a gender neutral possessive pronoun taking the place of any third person singular or third person plural pronoun (él, ella, usted, ellos, ellas or ustedes). Without context, you can often just pick any of the possible translations. I believe "its" is the simplest choice for "El gato bebe su leche." The cat most likely thinks of the milk as his.
The "tips" section of the home page for "Possessives" explains their usage fairly well.
So 'su' can be reflexive also? I know in some languages (e.g. Swedish) there is a different possessive pronoun depending on whether a reflexive sense is desired. I wonder how in Spanish we can force the reflexive reading, i.e., how can we say "The cat drinks it's own milk" in Spanish?