Can I unshare my real name on Battle.net?

It's not possible to disable it for just one person. However, you can disable it for all people.

There is a catch, though. Battle.Net / Blizzard may remove any friends who added you via email address. Personally, I don't think it removed any of my contacts when I did this.

If you still want to do this, in the Blizzard client:

1. Click the Blizzard icon.
2. Click Account Management.
3. On the new page, click the Settings menu then Communication Preferences.
4. Uncheck Enable Real ID.
5. There may be prompts that come up warning you about how Real ID friends may be removed.
6. Click the Submit button.

Having said that, many of Blizzard's games have the option of hiding RealID names in-game, although the streamer would have to do this, not you.

To disable the display of Real ID names in Overwatch:

1. Click Options
2. Click Social
3. Change "Display Real IDs For Friends" to "No".