I will try to satisfy my own conundrum my own way:
What is that entity that is above all? Likely no one has seen but most find it somewhere near them that we turn to? Its the God, the ईश्वर, the energy that runs the whole World including what we have not been able to see and above all that impacts us, our lives every day every second. No matter which religion it tries to explain, it tries to find who this entity is.
Then comes what in an umbrella word I would call Channels. This channel could be Knowledge (ज्ञान), Devotion (भक्ति), Practice (साधना) or any other Yog or way. We use these to seek the God or try to explain who this God is?
Then comes the Manifestation (यही है). We derive a manifestation of that energy in to something - This could have no form (निराकार) or have a form (साकार). This यही है creates school of thoughts, religions, traditions and impacts cultures.
What happens after Manifestation is what impacts us, our behavior, our practices our beliefs.
After Manifestation comes Traditions and then comes places of Worship like a Temple. Temples could be of a few types - One that are believed to be जागृत (awake) where ईश्वर का अंश जागृत अवस्था में हो, Second - those that are constructed to assist Humans in Worship or ज्ञान or simply as cultural centers. The temples usually have Idols after their प्राण प्रतिष्ठा - A practice or belief to create an entity that is awakened.
After Temple comes the Society that they serve. Temples make sure the society is together, is on the right path and enables it to keep the beliefs persistent beyond a human life. Temples also serve as places that enables or gives humans an opportunity to introspect, to seek with in, help them gather their thoughts or associate what ever other asset valuable to them with the ईश्वर.
After society comes the Manners (आचरण) - the human values, the Rights and the Wrongs. Temples and Traditions and Practices keep the Human values on the right path.
Then comes Us the Humans. After the Manifestation of the God each layer of this chain impacts us morally and ethically. Prior to the यही है is completely independent has no impact on what we do. We may get judged at some point by this Power but our own behavior would not change the Ultimate Truth.
Now back to the original question of हम शिव जी को दूध क्यों चढ़ाते हैं? Why do we offer Milk to Bhagwan Shiv at Temples?
First the Milk offered on Shiv Ji is not to waste it but it is to be distributed to all as a प्रसाद. It is often encouraged directly or indirectly for a भक्त to offer as much as he or she can afford. Why? Because Temples have responsibilities it is directly connected to a society that it runs in. A society is not equal to all even if the God is. The प्रसाद distribution makes sure every one gets a share equally even Milk that not every one could afford.
Second Milk is the most precious of all food. When a Baby is born it does not ask for Gold or Power, it asks for Milk. Milk is जीवन दायनी - A life source that could end a life without. Offering Milk is a way to associate our most precious thing as a living Human being to the God (अर्पण).
Third - Our Hindu Manifestation of God believes Shiv to have consumed Poison considered burning to protect the World from Evils. Milk considered to be implicitly cold is to pray Shiv and seek his Blessings.
Fourth - Milk is considered the Purest form of food purest form of Liquid. Remember after Sikhs considered an Army operation inside Golden Temple as objectionable they cleaned and washed the entire temple with Milk? Do you know Elders blessing "दूधो नहाओ"? Milk represents our supreme devotion (भक्ति) that we offer.
The Practice of offering Milk to Shiv Ji is a sign of Devotion to an entity Manifested into शिवलिंग रूप to be equally distributed among all believers of the faith the Temple is built to protect, preserve and spread.