Why do iMessage and FaceTime continue to charge money even after activation?

The most logical answer to this is that it is not sending international SMS messages after activation is done, so it must be something else your carrier is charging you for.

I would contact your carrier to get an explanation of the costs.

As far as I understand, many carriers have business relationsships with Apple and ensure that you are not charged for FaceTime/iMessage activation. You can check a list of carriers here:

https://support.apple.com/en-us/HT204039

UPDATE

You have updated the question with the requested information.

The phone number you quote is indeed an Apple registration number in the UK. Your own phone number seem to be in India. The carriers I know in Europe usually do not charge their customers for Apple registrations, but your Indian carrier apparently does.

The text message you have is not the FaceTime registration message - it doesn't look like that. It does however look like a standard receipt that indicates that a has been delivered succesfully.

I would try disabling and enabling FaceTime/Messages (i.e. specifically your phone number from FaceTime/Messages) to try to trigger a re-registration.

If that doesn't work, contact Apple to work out why it keeps trying to register.