Unexpected Continuity Camera behavior on Mac mini without built-in webcam

Luca, please report your suggestion to www.apple.com/feedback. Apple does not read or participate in the user to user forums which means all of us are end user just like yourself. You also may find another work around if you call AppleCare and speak to a technician. You can find the phone number for your country by clicking AppleCare Contact Info, selecting the country you reside in and then the phone number will appear.

You also may want to use Apple summarization functionality (part of AI) to summarize your thoughts if you run out of space. This is explained in more detail in:

"MacOS has a built-in summarizing feature, primarily powered by Apple Intelligence, that can create summaries of text, webpages, and even audio recordings. It can condense lengthy articles, key information, or lengthy notes into concise summaries, key points, lists, or tables. This feature can be accessed through context menus, the Safari app, or the Notes app. 

Here's a breakdown of how it works:

Text Summarization:

• You can select text, right-click, and choose "Summarize" to get a concise version. You can also use Writing Tools in the context menu to summarize, extract key points, create lists, or generate tables. 
• Webpage Summarization:
• Safari allows you to summarize webpages by clicking "Show Reader" and then "Summarize". 
• Audio Recording Summarization:
• In the Notes app, you can record audio and then generate a summary of the transcript. 
• Notification Summarization:
• You can customize your notification settings to get summaries of notifications, reducing interruptions. 
• Availability:
• Apple Intelligence is required, and it's available on Macs with M1 or later and macOS 15.1 or later, with specific language and region requirements. "

If you have a problem, then simply pick up the phone and call AppleCare and they will help you!! AppleCare Contact Info

"Unexpected Continuity Camera behavior on Mac mini without built-in webcam"

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See the Existing Cameras:

First off, see which, if any, cameras are even seen: 

1. Hold Down: the Option key
2. Click: the Apple menu
3. Choose: System Information or System Report
4. Select: Camera from the list on the left
5. Look For: any internal or external camera

"Hi, if i follow your path, the system information tells me that the mac mini does not have any video capture device. I think that the problem is exactly this, if i had a built-in video capture device, already installed like for example Mac book pro then continuity camera would have worked correctly"

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Continuity Camera:

Well, if you're not going to have a camera without one built in. So, is this seen when you boot with it plugged in? Save what need be, and then pour off your Mac. Unplug everything from it, and then plug it all back in. See if this external camera then shows.

Go here: Continuity Camera: Use iPhone as a Webcam for Mac - Apple Support

• Mount your iPhone
• Choose your iPhone camera or mic
• Use Reactions, Presenter Overlay, and other effects
• Pause or disconnect
• Turn off Continuity Camera
• If you need help
• System requirements

"Unexpected Continuity Camera behavior on Mac mini without built-in webcam"

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For Tip:

If you want privacy, you can turn off your internal camera, through use of a profile. If you not be used, and only your Phone's camera would potentially be seen. Here is my User Tip: How to Use a Profile to Disable the iSight Camera in macOS 15 Sequoia - Apple Community