Again, it seems WB isn't allowed to use "MGM" or "Metro-Goldwyn-Mayer" branding on their products because MGM is still a registered trademark. That's why many of there MGM sets use more roundabout ways for advertising them, like the "Dream Factory" collections and so-forth.
Throughout most of the '90s, Warner Brothers was allowed to use the MGM branding and trademark, such as when they went about making a documentary miniseries called MGM: When the Lion Roared, because Warner Brothers had struck a deal with MGM/UA that allowed MGM's home video department to release the WB-owned MGM classics, which lasted up until 1999.
As to why they don't work together anymore, my guess is that MGM's current arrangements with 20th Century Fox make it a sticky proposition.
Fox was playing catch-up to Warner. Warner released far more classic titles from 2007-2012. But let's just ignore everything they released prior to 2013 for a convenient yet incomplete narrative...
And let's not be too quick to forget Fox's 75th anniversary in 2010. Compared to the other anniversaries that have popped up since, it was one big dud.