"could not connect" to Apple TV.

Sorry, I can’t think of other reasons that might prevent AirPlay from working for one Apple TV and one notebook only.

As that last line on the linked support document says:

If you still can’t use AirPlay after trying these steps, contact Apple Support.

Give them a call, or start your support request online.

If others are connecting when you are not able to, then the suspicion is with the configuration of your MBP.

- Do you have any third party software running that might interfere? “Security”, “Cleaners”, “Anti-Virus”, Network monitoring/​firewall, etc.

- Can you AirPlay to other Apple TV units on that network, if any?

- Verify with the network administrator that there aren’t any restrictions based on device identification (e.g. MAC address). And if there are, then add your MBP to the ‘Allowed’ list.

If AirPlay or screen mirroring isn’t working on your device - Apple Support