This seems incredibly odd to me.
On one hand, they indicate strongly valued privacy with the "no posts about the child" rule; seemingly recognizing the hidden costs of permissive 'oversharing'.
But, on the other hand, they place a very high importance on reserving (vanity) domain and user names, securing the right for her to sharecrop out her personal information in as-accurate-a-means as possible; reinforcing the value and importance of being able to overshare?
e.g. Reserving JaneAnnSmith.com at birth to ... what? Spare her the inconvenience/indirection of someone else buying that domain in the intervening years and her having to instead register jasmith.com or (heaven forbid) jane-ann-smith.com ?
Securing facebook.com/janeannsmith to spare her facebook.com/123456789 or something?
And does anyone even think for a moment that it's likely for today's real-name-based social networks will be relevant, years from now?
Reserving a Facebook name today is akin to having reserved a geocities name, at birth, for a child who will graduate high school next year -- and that's for the earliest of early adopters. Were the name registered for a child born around geocities peak, that child may still be in junior high.
The ephemeral nature of these networks (over such time frames) only further underscores the apparent importance the parents are placing on precisely identifying her online history. (getting that facebook username just in case it's still relevant...)
It seems rather than teaching their child to value privacy, to understand the power/benefit dynamic of these networks, to hopefully avoid her growing up forever linked to juvenile mis-steps, they're instead raising her to be a narcissist.
That is: Not to value her privacy and personal identity in and of themselves, but to value privacy and personal identity as a tool to 'spin' her public identity in the most-favorable fashion.