Dictionary.com says of the idiom
milk of human kindness, the
Compassion, sympathy, as in There's no milk of human kindness in that girl—she's totally selfish. This expression was invented by Shakespeare in Macbeth (1:5), where Lady Macbeth complains that her husband “is too full of the milk of human kindness” to kill his rivals.
Here's an interesting fact, in Czech the expression krev a mlíko is translated in English as milk and blood; however, its meaning is far removed from that of Shakespeare's milk of human kindness. It refers to a voluptuous curvy woman, it's also worth noting that milk is clearly associated with the fairer sex, and despite the idea of it combined with blood stirring in me
a sense of mild disgust, "milk and blood" does suggest someone who is expressly feminine yet passionate, and added bonus, lusty too. Lest we forget, blood is red; the color of love and passion.
Similarly, the themes of blood and milk often make their appearance in Macbeth, as @TaliesinMerlin's answer also illustrates.
blood
In the famous line, Lady Macbeth accuses her husband of being compassionate, of being too nurturing, of acting motherly. Today, some might use the slang “wuss”: failing to do or complete something as a result of fear or lack of confidence, this accusation could have been aimed at a person who had never entered into battle but to a Scottish general? Macbeth, by all accounts, was a valiant warrior whose sword had slashed and massacred innumerable lives.
"For brave Macbeth — well he deserves that name —
Disdaining Fortune, with his brandish'd steel,
Which smoked with bloody execution,
Like valour's minion carved out his passage
Till he faced the slave." I. ii. 16-20.
and
"Bellona's bridegroom, lapp'd in proof."
I. ii. 54.
Bellona, the Roman goddess of war, and Macbeth her groom.
milk
Yet, Macbeth's very own wife, the person who knew him better than any other accused him of being a loving nurturing mother. “Human milk” i.e. breast milk, is naturally warm and fed to babies, the most vulnerable and innocent participants of mankind.
The imagery of masculine resolve and unbridled ambition is later summoned by Lady Macbeth
I have given suck, and know
How tender 'tis to love the babe that milks me:
I would, while it was smiling in my face,
Have pluck'd my nipple from his boneless gums,
And dash'd the brains out, had I so sworn as you
Have done to this.
I suggest that Shakespeare uses the expression “milk of human kindness” in its literal sense, milk was known to be nourishing and essential for babies to prosper and grow but it was also believed to contain human personality traits. Elizabethans knew that breastfeeding was responsible for creating the profound bond between a mother and her child.
“In the fifteenth century, as Paulus Bagellardus tells us, a
-mother customarily suckled her child for two or three years.”
Due in part to this bonding period, several superstitions about breast milk persisted throughout the 15th and 16th century. Wet-nurses were employed to breast feed babies whose biological mothers were unable to suckle but it was essential that these women were of good character lest the infant should suck in the vices of a sluttish or evil-tempered wet-nurse.
… for example, was the ancient belief in the transmission of qualities of temperament from nurse to child. In solemn proof whereof we are given the story of the Queen of France who, solicitous of the welfare of her infant, devoted both time and energy to suckle him herself. One day, however, she surprised a great lady of the court giving the babe her breast. Horrified and indignant, the Queen thrust her finger down the child's throat, and he, perceiving (so it almost seemed) the danger to which he had so rashly exposed himself, rid his stomach of the illicit meal, thus preserving his royal humours from contamination with merely aristocratic virtues and providing an, object-lesson to all and sundry of lower degree who might look too lightly on wet-nursing.
Source: The History of Infant Feeding from Elizabethan Times.

A poor woman in a dingy attic, surrounded by her children, one of whom she is breast-feeding.
Engraving by N. de Larmessin III after J. Pierre.
This superstition is also confirmed by Dr Victoria Sparey
As a substance of the humoral body, breast milk was accordingly understood to shape the physicality and mentality of the suckling child. As Thomas Raynalde observed, ‘affections and qualities [of the nurse] passeth forth through the milke into the child, making the child of like condition and manners.’
But Raynalde also argued that the milk produced by the mother was more
agreeable and nourishing for an infant, noting that it was the same blood they consumed in utero, only converted to milk.
I am of the opinion that it is fit for every Mother to nurse her own Child, because her milk which is nothing but the blood whitened, which nourished the child in the womb and of which the child was conceived…
The 'byrth of mankynde' Thomas Raynolde (1540)
Lastly from the Old Testament, we have the binomial pair milk and honey. From the King James Bible, 1604-1611
- Thy lips, O my spouse, drop as the honeycomb: honey and milk are under thy tongue; and the smell of thy garments is like the smell of Lebanon.
Source: Song of Solomon 4:11
- And he hath brought us into this place, and hath given us this land, even a land that floweth with milk and honey
source: Deuteronomy 26:9
For over 1,500 years milk had represented prosperity and goodness, its positive symbolism had been firmly established long before Shakespeare had penned the line "Yet do I fear thy nature; it is too full o' the milk of human kindness."