Well, spanky, I figured Hinduism would've forbidden milk due to it being an animal product, especially from a cow, which they consider sacred, right?
Hinduism doesn't forbid 'animal products'. In fact, the only thing 'It' forbids generally is cow flesh. But the sects of Hinduism that forbid meat eating in general do this have specific reasons.
I grew up as a Gaudiya Vaishnava. The three predominant reasons for not eating meat were:
1) Animals are sentient and killing them causes suffering to them. Their life is also valuable, and we would be taking it away from them by killing them
2) Everything is embodied by a certain mode of nature. These are goodness, passion and ignorance. So when we eat something, it will have a particular affect on our body, mind and consciousness. Food that is in the mode of goodness will be healthy all around. Food in the mode of passion will give us energy, make us more passionate. Food in the mode of ignorance is generally negative.
Meat is considered to be in the mode of passion. High levels of energy (often nervous energy) are not recommended for a spiritual lifestyle, which requires a clam mind and the ability to focus and stay quiet/calm for certain lengths of time (such as during meditation).
3) Karma. If you cause suffering, that suffering will come back to you. The more karma we accumulate, the longer we will have to remain on the material platform of life. It is a vicious cycle.
So to answer your question, Milk is considered in the mode of goodness. Whereas eggs are not. And because the Cow is one of the sacred Mothers (according to Hinduism), her milk is like taking food from a mother. And Hindus are very grateful and loving about it. (As a side note, the eating of cow flesh is considered so repulsive for the same reason- it's like eating your own mother)