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:
- Click the Blizzard icon.
- Click Account Management.
- On the new page, click the Settings menu then Communication Preferences.
- Uncheck Enable Real ID.
- There may be prompts that come up warning you about how Real ID friends may be removed.
- 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:
- Click Options
- Click Social
- Change "Display Real IDs For Friends" to "No".