The comparison of her lips to a scarlet thread is probably focusing on the colour of her lips.
The dominant colour in Song of Solomon 4:3–4 is red. Scarlet dye was a luxury substance, and the colour was associated with life and with love.1 The Hebrew word used for mouth (midbar) is unusual, focusing on the mouth as the organ of speech. It is not merely the appearance of her mouth that is lovely, but the words she speaks too.2
3 Your lips are like a scarlet thread, and your mouth is lovely. Your cheeks are like halves of a pomegranate behind your veil.