Grate the zucchini and place the shreds on a clean kitchen towel. Cover the zucchini with the towel and press on it to squeeze the moisture out. Leave the shreds on the towel for now to absorb any extra moisture.
Position an oven rack to the center of the oven and preheat the oven to 350°F. Grease a 9x5 loaf pan or line it with parchment paper.
Mash the bananas in a large bowl with a fork. Add the zucchini, brown sugar, plant milk, oil, and vanilla to the large bowl. Stir these ingredients together until they're well combined.
Use a sifter to add the remaining ingredients except for the chocolate chips. Fold them together till they're just combined. Try not to overmix so the bread doesn't become tough and chewy. Gently stir in the chocolate chips.
Spoon the thick batter into the loaf pan. Smooth the top of it with a spatula or metal spoon. Give the loaf pan a few taps on the counter to help the batter settle into the grooves. Top the bread with more chocolate chips.
Bake the zucchini banana bread for 50-60 minutes in the center of the oven. It's done when a toothpick is inserted into the center of the bread comes out clean. It's okay if a few moist crumbs or melted chocolate are on the toothpick.
Let the bread cool for 10 minutes in the loaf pan. Then, remove the bread from the pan and transfer it to a rack to cool completely before slicing it.