1) Most important:
If you buy for other people, your name then becomes tied to their posts. If father Bloggs buys Bloggs Jr SC2 and Bloggs Jr then posts "I hate Antarcticans", people (employers, potential friends) might think Bloggs Sr is racist.
This also applies to stolen cards, and people other than children for who you may have responsibility.
There is potential for Blizzard to be sued over this.
2a) Trolls are often dedicated. If a troll needs to fake a name, and it isn't too hard to do, s/he will do so and continue trolling.
2b) To prevent this, you will now be required to supply ID on purchase. To prevent trolls faking, and to make it global, the only option is to require all purchases supply credit card info as there is no other global ID system.
2c) If you required national ID in some countries, trolls could just buy on-line from other countries via proxy.
2d) Game shops don't have the facilities & training to detect fake IDs, and they won't be happy being told they need to pay for them.
Even this in no way prevents people buying the game for other people.
3) New system requirement: Credit (or debit) Card.
I see this as Blizzard trying to do the impossible. Like DRM, this will cause problems for legit people that value their privacy but won't stop those that it intends to: trolls.