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 Tes lèvres sont comme un fil cramoisi, Et ta bouche est charmante; Ta joue est comme une moitié de grenade, Derrière ton voile.