Can a bow get a non-bow enchantment?

Short answer: No

Long answer: No, unless you have a mod.

Yes. If you make an Enchanted Book you can add a non-bow enchantment. If you are in Creative mode, you can enchant a bow with any enchantment; if you are in Survival mode, however, you may only enchant it with Unbreaking.

Here is an easy way to do it on Creative mode, or Survival mode. (No Mod's Required)

1: You get an anvil, and in the first slot put the bow, and in the second slot put the enchanted book of looting there. Then you should have a bow enchanted with looting now. (The Image is from my Minecraft.)