They did, but the name given by Pharaoh’s daughter was foremost. Allow me to explain.
We are taught that Moshe actually had 10 names. His father called him Chever, his mother called him Yekutiel, his sister Miriam called him Yered, etc.
Yet the name given by Pharaoh's daughter was the one chosen by God. The Torah never refers to him by any name other than Moshe. Why?
Pharaoh's daughter saved Moshe's life and adopted him and cared for him as her very own son. Therefore, she merited that her name prevailed. Moshe himself may have used this name out of gratitude to her.
Another reason the Torah calls him Moshe is the significance of the name itself. The name "Moshe" means that just as he was rescued and drawn from the water, so too he will rescue others from hardship, and that is what he did.