The JZ4780 has an internal real-time clock (RTC) peripheral which maintains time and also provides control over hibernation and power-down operations. See the drivers/rtc/rtc-js4740.c driver in the Linux kernel. The Power information describes the interaction between the RTC peripheral and power operations.
The Linux kernel driver attempts to access the RTC peripheral registers and tests the scratchpad register for a value that it has previously set. Where this fails, the driver seems to conclude that it is not able to access the registers. In fact, any differing value merely indicates that the RTC has not been able to keep the time. With a secondary RTC chip in place, the driver could reset the peripheral with any available time information instead.
Since the RTC peripheral provides registers for orchestrating power-down, the driver should register itself in some sense, anyway. Otherwise, power-down functionality is no longer available.
According to the online documentation, this is a Pericom PT7C4337 which seems somewhat compatible with the Dallas DS1337. See the drivers/rtc/rtc-ds1307.c driver in the Linux kernel.
However, the V2a board seems to employ an AT8563S which is compatible with the NXP PCF8563 and is supported by the drivers/rtc/rtc-pcf8563.c driver in the Linux kernel. It may even be the case that this chip is used on the first board versions and the above Pericom information is incorrect.