Here's the scenario. I have two 2013 test accounts setup in Outlook as POP/SMTP accounts. I have their name setup as initials in the outlook account (the "Your Name" field). If I send an email (Through SMTP) to a 2013 Exchange(Outlook Anywhere) client, their names resolve to their actual Display Name in the GAL which is correct. Now if I send an email to another 2013 POP/IMAP user, their name shows what is setup in Outlook which is their initials which is not correct.
Now we are currently in the process of a 2007 to 2013 migration. I do not see this behavior on the Exchange 2007 side with POP/IMAP clients. If I run the same test setup with a 2007 mailbox as the recipient, the sender's name resolves to their GAL display name and NOT the name configured in the Outlook Account settings. It doesn't matter whether the sender is a 2007 or 2013 account. As long as the recipient is on 2007 it works fine.
This is all tested using authenticated SMTP. The only difference I can tell is that the 2007 servers resolve the name before delivering it to the inbox where the 2013 servers don't.
There aren't any GAL or Address Book related issues on the 2013 side. Full exchange clients or OWA work correctly.