The MyLinks control is hidden in SharePoint 2010, but as you point out still present in both 2010 and 2013. The links are part of the User Profile. So they are in the User Profile Database. You can access them via the QuickLinksManager and the User Profile Property "QuickLinks".
I should point out that the preferred technique in 2010 and 2013 is to use Tags rather than quick Links and there is a feature "Create Tag from Link" to assist in this transition.
$context = Get-SPServiceContext -Site http://mysite
$upm = New-Object Microsoft.Office.Server.UserProfiles.UserProfileManager($context)
$profile = $upm.GetUserProfile("dht\ruby")
#Get the QuickLinks Manager
$qlm = $profile.QuickLinks
$qlm.GetItems() | fl