When Nokia announced the Amber update to Windows Phone they noted it will bring FM Radio support to all their handsets except the Nokia Lumia 620, which apparently lacked the FM Chip.
WindowsMania.pl, who has schematics for the handset, the RM 846, notes that the device has exactly the same dual-core chipset as all the other low-cost Windows Phone 8 handsets, the MSM8227, and that the FM Radio is integrated into the WIFI and Bluetooth Chipset.
The schematics further show an FM Antenna is also connected to the chipset, meaning theoretically it should work.
A more likely reason, suggested by psychomania over that Nokia Discussions, is that the Nokia Lumia 620 has a rather unique arrangement where the headphone jack is part of the removable shell, which may prevent the headphone jack working well as an FM Antenna.
This reason would of course be more difficult to explain, but I suspect a more truthful and accurate answer is always worth the time explaining, as the real story will always eventually out in the end.
(Headphone jack picture by allaboutwindowsphone.com)