Indeed. According to the Midrash (Genesis Rabba: Hayei Sarah: 60:31) Eliezer is not just a Canaanite, but the Canaanite; Canaan himself!
ויאמר בא ברוך י"י... אמר ר' יוסי בר' דוסה כנען הוא אליעזר ועל ידי
ששירת את הצדיק באמנה יצא מכלל ארור לכלל ברוך שנ' ויאמר בוא ברוך י"י
It should be noted that his identity as a Canaanite is a Midrash (Genesis Rabba 59:9), and the story about his daughter is a Midrash, So the latter doesn't really support the former. If one accepts the Midrashim as independent proof, then the latter is not needed, and if not, the the latter doesn't help.
Although many Midrashim make this identification, none of them state what pushes them to make this identification, nor have I found any later commentators who explain the association. It should be noted that Midrashim interpret דמשק (literally Damascene) non-literally (e.g. Yoma 28b, Genesis Rabba 44:9, Midrash Agada (Buber) 15:2), which is perhaps consistent with the Midrashic identification of Eliezer with Canaan.