Friend Owl: You know, just the other day, I was talking to myself about you, and we were wondering what had become of you.
Young Bambi: He stopped and looked at me.
Bambi's Mother: Yes, I know.
Young Bambi: Why was everyone still when he came on the meadow?
Bambi's Mother: Everyone respects him. For of all the deer in the forest, not one has lived half so long. He's very brave and very wise. That's why he's known as the Great Prince of the Forest.
Thumper: Whatcha gonna call 'im?
Bambi's Mother: Well, I think I'll call him Bambi.
Thumper: Bambi... Yep, I guess that'll do all right.
Young Bambi: What happened, Mother? Why did we all run?
Bambi's Mother: Man was in the forest.
Thumper: "Eating greens is a special treat, It makes long ears and great big feet. But it sure is awful stuff to eat." I made that last part up myself.
Answer: Probably so he would not think he is superior to the other fawns and try to lord over them. She would have told him about his father when he was older. This is also a literary plot device by setting up a dramatic reveal at a crucial time (right after Bambi's mother is killed) that Bambi learns his father is the Great Prince. This has a greater emotional punch.