I have in my library Baldur's Gate II Complete, which I see has been superceded by Baldur's Gate II Enhanced Edition in the store.

Do I have to buy the Enhanced Edition? Or is there some way that I get the Enhanced edition for free because I have the Complete edition?

I think that if you could get the Enhanced Edition free, it would already be in your library, without you needing to take any action. I know for some games the upgraded version is automatically gifted to people who had the original version, but I think in the case of Baldur's Gate, that was not done. There may have been a time-limited sale where you could get the EE for a discount, but if there was, that sale is long gone. The games go on sale at a substantial discount during the quarterly sales, so if you want EE, and are willing to wait for it, you could get 50%+ off pretty easily. I think I have seen better than that on occasion.

If you buy the Enhanced Edition, you get a gift code that lets you get another copy of the original edition. That only helps you if there's someone you want to give it to though.

You don't "have" to buy it. You could if you wanted to experience the added content, which is mediocre at best, and also smoother mutiplayer, or at least easier to set up. The rest of the enhancements are 80% based on the same mods made by the community for bg2. The rest is the added npcs and The Black Pits 2, which is just a single player arena mode.