This is a kind of bug report.
When I connect from Discord to Twitch,
Discord didn't appear in Twitch connections page.
https://www.twitch.tv/settings/connections
As a result, when I'm streaming in twitch, my Discord status doesn't turn into "Streaming".
Of course I enabled streamer mode.

This issue seems to occur for seven years.
https://www.reddit.com/r/discordapp/comments/4jv325/status_doesnt_turn_purple_when_streaming/
One of my friends also failed, but the others succeeded.

I solved this by repeating reconnecting until Discord appeared in Twitch connections page.

The connection processing should be reviewed.
This issue is not from Discord to Twitch, but from Twitch to Discord.