Ambereen means "like anber". It is formed from Farsi and Arabic Anber (which is commonly pronounced as Amber), which refers to a very expensive material taken from whales (known as Ambergris in English) that in the past was used in the making of perfume, and Farsi een ("like", "full of"). The full meaning is "extremely good scented", "like a wonderful perfume".
When used in describing a person's eyes, it means extremely black, like the color of Ambergris.
Since the meaning is good, it is an acceptable name for Muslims. It seems to be mostly used for girls.