Xperia Z2: SD Card not being shown as mounted under Storage, but is accessible from file manager
After a lot of research and browsing the XDA-Developers Forum, it appears that the problem lies within the stock kernel I'm using, so I flashed this custom kernel instead http://forum.xda-developers.com/xperia-z2/development/kernel-andropluskernel-v10-t2974841 which appears to solve the issue.
I rebooted my phone a few times, and my SD card is being mounted properly after each reboot.
A word of caution though, don't flash this kernel if you're using XZDualRecovery, you must uninstall it first (there are detailed steps in the post I linked above).