The phone does not recognize the Apple TV

Apple TV (4th generation or later) should automatically get recognized through iCloud and set up as HomeKit home hub (‘connected’, or possibly ‘standby’ if there are multiple hub devices). It doesn’t need manual adding. It does need to have the same Apple ID account for iCloud services. It must match the account for iCloud services as used in the Home app on iPhone (or iPad).

Set up your HomePod, HomePod mini, Apple TV, or iPad as a home hub - Apple Support

Also see: Home app: (⋯) Home Settings﹥Home Hubs & Bridges﹥[Apple TV name]﹥Connected.

… when you bring the iPhone close to the box …

Pair your iOS device or Mac with your Apple TV - Apple Support

I configured the I CLOUD manually …

Not sure what you mean by that, but the Apple ID for iCloud services is what defines HomeKit. It should match between iPhone with the Home app and Apple TV box. Else that is the reason why there is no recognition.

Do you have the Apple TV box? What is its model and version system software? Identify your Apple TV model - Apple Support

What is your model iPhone and its version iOS? Identify your iPhone model - Apple Support

Find the software version on your iPhone, iPad, or iPod - Apple Support

HomeKit requires Apple TV (4th generation or later) for home hub functionality. Third party devices with the Apple TV app do not qualify for that. The Apple TV box itself is not a smart accessory for programming it what to do or when, nor used for managing HomeKit. The home hub just ensures a connection with iCloud. Managing the HomeKit functions and configuration is done in the Home app on iPhone or iPad.

The phone does not recognize the Apple TV

Where and what for did you expect it to recognize the Apple TV?

a dialog box does not appear on the phone

What dialog box did you expect? What were you trying to do?

I can't manage to configure the smart home

What HomeKit compatible smart home accessories do you have that need configuring?

Set up your HomePod, HomePod mini, Apple TV, or iPad as a home hub - Apple Support

Add an accessory to the Home app - Apple Support

Get notifications for your smart home accessories - Apple Support

Create scenes and automations with the Home app - Apple Support