Yes, doesn't that neutral possessor rule out cows and goats? I am not trying that milk. I mean which animal's milk is that? Do we really have to drink it? Unless the possessor is a well known dairy company which sells cow's or goat's milk, because then I would try it.
In Danish, they use the same words for possessive pronouns and adjectives like 'min' is both 'my' and 'mine'. When it comes to "Jeg" (I) and "Du" (You), the possessive has an extra pronoun/adjective if what you own is in plural.
For example, if you own an apple, "æble", which is an -et word, you would say, "Æblet er mit" (The apple is mine) or "Mit æble" (My apple) , and if you own many apples, you would say, "Æblerne er mine" (The apples are mine) or "Mine æbler" (My apples).
If what you own is an -en word, like kat then it would be, "Katten er min" (The cat is mine) and "Kattene er mine" (The cats are mine).
For the second person (You) those sentences would be: "Æblet er dit" or "Dit æble" and "Æblerne er dine" or "Dine æbler". Then, "Katten er din" or "Din kat" and "Kattene er dine" or "Dine katte".
See the correspondence of -it ending possessives with -et ending words? :) Just as -in ending possessives with -en words.
For "his", "her/s", "our/s", plural "your/s" and "their/s" it's all the same, "hans", "hendes", "vores", "jeres" and "deres" as you might have noticed already.
Then, if you're wondering about when to use "sin, sit and sine". It's just that Danes really want to make sure you tell them who owns something when you say, "It's his/her [enter thing here]..."
SO to exemplify this, I'm going to ask you to imagine yourself talking to a friend while two other people are playing with a child and a dog.
If you wanted to tell your friend that the guy is playing with the girl's dog, you would say, "Han leger med hendes hund" (He is playing with her dog), but when the girl plays with her own dog, it would be, "Hun leger med sin hund" (She is playing with her [own] dog).
And when the girl is playing with the guy's child, you would say, "Hun leger med hans barn" (She is playing with his child), but when the guy is playing with his own child, then it would be "Han leger med sit barn" (He is playing with his [own] child).
The only difference you have to remember now is that for third person impersonal ("it", the pronoun not the clown :p ) the possessive pronoun changes according to who owns the object, and not the object itself (as with the previous ones).
In this case, even if a dog owned an apple, it would still be "Dens æble" (Its apple). Why? Because hund is an -en word and whoever is doing the owning is who we care about when it comes to the possessive pronouns for "it".
In the same way, if some animal (dyr, -et word) had a cat (kat, -en word) as a pet, then you would say, "Dets kat" (Its cat) because the animal is the one owning the cat.
Sorry for the super long reply! I wasn't sure what exactly it was you didn't understand about possessive pronouns, so I wrote down everything I understand so far. If there's something wrong, I hope someone corrects me, but it's all worked for me so far, so I hope it helps you too :)