Microsoft now acknowledge there's a problem for people who registered to Xbox LIVE under the age of 18 (Child Account with parental consent) but reached the age of 18 by now. These accounts are still Child Accounts and thus cannot access the Messenger feature in the new dash. There's no solution for now.
I am now 18 or older. Why is my account still considered a child account?
There is a very small group of Xbox LIVE members who first created their accounts while under the age of 18 but have since passed their 18th birthday. These members cannot access Messenger on Xbox 360 at this time. We will explore solutions that enable these members to use Messenger on their Xbox 360 and will update the community when we have more information to share.