Your iPhone might show a "DND" status to others even when Do Not Disturb isn't active if you're using Focus modes and have enabled "Share Focus Status" for certain Focus modes, which can be turned off in Settings > Focus > Focus Status.
Here's a more detailed explanation:
Focus Modes and "Share Focus Status":
The iPhone uses Focus modes (like Do Not Disturb, Sleep, Driving, etc.) to manage notifications and other features. You can enable "Share Focus Status" for specific Focus modes, which allows apps you've granted permission to share that you're using a particular Focus mode (and therefore silencing notifications).
How It Works:
When you have "Share Focus Status" enabled for a Focus mode, apps can see that you've silenced notifications, and people trying to send you a message will see that you've silenced notifications.
Turning Off Status Sharing:
If you don't want to share your Focus status, you can disable it in Settings > Focus > Focus Status. You can either turn off "Share Focus Status" for all Focus modes or customize which Focus modes are able to share your status.
Do Not Disturb as a Focus Mode:
Do Not Disturb is a Focus mode, and if you have "Share Focus Status" enabled for it, then when Do Not Disturb is active, your status will be shared.