In this chapter Solomon is not talking about a favorite female, he is reminiscing on the beauty of the Tabernacle structure, and using it as a reference to construct the new temple. Solomon's Temple would be legendary for its glory and construction. Here is a link to Chapter 4 so you can read along and it will make sense.
In verse 3 we find the interesting phrase "your temples behind your veil." Using the word "temple" and "veil" in the same sentence should perk us up as to the hidden meaning. A little earlier Solomon phrases it slightly different: "you have dove's eyes behind your veil." So what does it mean? Solomon is praising the beauty of the Holy of Holies behind the temple veil. Priests had to be very careful in this area as making a mistake here could be fatal as Uzziah would learn the hard way. This is the place God's very presence would rest, over the mercy seat upon the ark of the covenant. And so the "doves' eyes" is the presence of the Holy Spirit who lives behind the veil in the temple. The dove is a primary biblical symbol of the Holy Spirit! When Jesus was baptised, scripture says the Holy Spirit rested on Jesus in a form like a dove: Luke 3:22 And the Holy Spirit descended in bodily form like a dove upon Him, and a voice came from heaven which said, "You are My beloved Son; in You I am well pleased."
goat's hair
Verse 1, Your hair is like a flock of goats, going down from Mount Gilead.
In saying "Your hair is like a flock of goats" Solomon is recalling the tabernacle covering:
Exodus 26:7 You shall also make curtains of goats' hair, to be a tent over the tabernacle
"shorn sheep"
In verse 2, Solomon lists "shorn sheep." This is another ingredient as the tabernacle required a covering from the skins of male sheep:
Exodus 26:14 You shall also make a covering of ram skins dyed red for the tent,
"a strand of scarlet"
Verse 3, "Your lips are like a strand of scarlet"
Exodus 26:1 "Moreover you shall make the tabernacle with ten curtains of fine woven linen and blue, purple, and scarlet thread"
"pomegranate"
v3 "Your temples behind your veil are like a piece of pomegranate"
Pomegranates were prominent in the design of Solomon's temple:
1 Kings 7:42 "Four hundred pomegranates for the two networks (two rows of pomegranates for each network, to cover the two bowl-shaped capitals that were on top of the pillars);"
"shields of mighty men"
Verse 4, Your neck is like the tower of David, built for an armory, on which hang a thousand bucklers, all shields of mighty men.
Like the pomegranates mentioned previously, the shields are a part of Solomon's temple. This is a similar but different structure than the tabernacle of Moses. In the beginning verses Solomon was first recalling the tabernacle so that he could use the understanding of what God desired in its blueprints when designing the new temple. Solomon's temple would be a permanent structure, signifying the permanent establishment of His promise. God's people were in the promised land to stay and God wanted to always dwell among them. The tabernacle was designed as a mobile structure which could be packed and moved when neccessary. Solomon's temple signified the permanent relationship of God with his people and of his people with their promised land.
1 Kings 10:16 And King Solomon made two hundred large shields of hammered gold; six hundred shekels of gold went into each shield. He also made three hundred shields of hammered gold; three hundred shekels of gold went into each shield."
These shields were originally placed in the house of the forest of Lebanon, but later moved to the Temple itself. Later on, in the time of Rehoboam, these gold shields were taken as plunder in an invasion by Egypt. In embarrassment, Rehoboam had bronze lookalikes manufactured and would have his soldiers array them at the temple only when he was to go there. As soon as the king left, those shields were taken back to a secure storage place. This story can be found in 1 Kings 14:25-28.
For verse 5 we already have an entire article on the breasts in Song of Solomon.
"with one link of your necklace"
Verse 9, You have ravished my heart with one look of your eyes, with one link of your necklace.
Exodus 28:22 You shall make chains for the breastplate at the end, like braided cords of pure gold.
The priest would wear the holy breastplate held by chains like a large pendant on a necklace. This is what Solomon is describing by the chain link necklace imagery.
Myrrh and Frankincense
Song of Solomon 4:6 I will go my way to the mountain of myrrh and to the hill of frankincense."
:14 Fragrant henna with spikenard, spikenard and saffron, calamus and cinnamon, with all trees of frankincense, myrrh and aloes, with all the chief spices."
Exodus 30:22-25 Moreover the Lord spoke to Moses, saying: "Also take for yourself quality spices-five hundred shekels of liquid myrrh, half as much sweet-smelling cinnamon (two hundred and fifty shekels), two hundred and fifty shekels of sweet-smelling cane, five hundred shekels of cassia, according to the shekel of the sanctuary, and a hin of olive oil. And you shall make from these a holy anointing oil."
:34-35, And the Lord said to Moses: "Take sweet spices, stacte and onycha and galbanum, and pure frankincense with these sweet spices; there shall be equal amounts of each. You shall make of these an incense, a compound according to the art of the perfumer, salted, pure, and holy.
Stacte is believed to have been drops of Myrrh. The 4th-3rd BC botanist Theophrastus wrote, "From the myrrh, when it is bruised flows an oil; it is in fact called "stacte" because it comes in drops slowly." And the 1st century writer Dioscordes, in his Materia Medica (Section 1.73,79), considered myrrh to be the source of stacte, and he describes Storax and Styrax separately.
The myrrh and frankincense Solomon mentions, foreshadows the sacrifice Christ would make as his sacrifice ended the old temple order once and for all and replace it with a beauty far better. We see these two substances offered to Christ by the wise men:
Matthew 2:11 "And when they had come into the house, they saw the young Child with Mary His mother, and fell down and worshiped Him. And when they had opened their treasures, they presented unto Him gifts: gold and frankincense and myrrh."
Solomon saw the past in the tabernacle of Moses, the present in the temple he was currently constructing, and the future in Christ's doing away with animal sacrifices and building a temple of His people in which we are "living stones." Comparing his favorite woman's hair to a goat's would be the opposite of a romantic phrasing, but when you see it as a vision of the tabernacle, it all clicks together.