Hi RN93,
Thank you for using the Apple Support Communities! You can turn on Share Focus Status as listed below, however, this is not customizable:
Use Focus on your iPhone or iPad
Share your Focus status
When you set up a Focus, turn on Share Focus Status so that apps display that you have notifications silenced when people message you. Since apps only know that you have notifications silenced but not which Focus you have turned on, the name is never shared. Focus status is shared in apps when you have a Focus turned on and after you give an app permission.
To turn Focus status on:
1. Go to Settings > Focus.
2. Tap Focus Status.
3. Turn on Share Focus Status.
4. Select which Focus options can share that you have notifications silenced.
Another feature is Auto-Reply when using the Driving Focus which you can customize:
Use the Driving Focus on your iPhone to concentrate on the road
1. Go to Settings > Focus, then tap Driving.
2. Tap Auto-Reply.
3. Select an option, like No One, Recents, Favorites, or All Contacts.
From here, you can customize the auto-reply message that people get when they message you while a Driving Focus is on.
Hope this helps!
Have a great day.