Hello nfilipek,
You can use Continuity Camera with these apps:
Use an app that supports Continuity Camera
Use Continuity Camera to scan or take a picture in these supported apps:
• Finder
• Keynote 8.2 or later
• Mail
• Messages
• Notes
• Numbers 5.2 or later
• Pages 7.2 or later
• TextEdit
You can find more information here: Continuity Camera: Use your iPhone or iPad to scan documents or take a picture on your Mac - Apple Support
If you're trying to use your iPhone as a webcam, try these steps:
Choose your iPhone camera
Open FaceTime or another app to use with your iPhone camera. If the app doesn't automatically begin using it, choose your iPhone from the app's video menu, camera menu, or other setting. For example, in FaceTime your iPhone camera is available from the Video menu.
Continuity Camera: Use iPhone as a webcam for Mac - Apple Support
Should the issue persist, contact Apple Support. They will be able to look into this with you further.
Have a great day!