I guess its through the diag port.
To clarify, it's via the BMW DS2 BUS connected to Pin 8 of the OBDII connector on which the instrument pack is also connected.
The instrument pack then transparently (Acting as a Gateway) relays diagnostic messages addressed to the AMP onto the I-BUS, on which the AMP (Along with other ECU's) is also connected. Replies return to the diagnostic tool by the opposite route.
Although the ECU's in use for systems on the pre 2005 L322 were derived from existing BMW ones, and they look and connect the same, and they may even have identical hardware inside, sadly in my experience, the Land Rover versions are not generally interchangeable with BMW ones. And re coding is certainly not typically a one fits all way to make them do so.
The dealer can code the amp back to work in the Range Rover enviroment
As you may appreciate, i have rather more than a little knowledge and many years of hands on experience with the LR dealer equipment, its capabilities and how it works. I can therefore categorically tell you that unless the HW number, SW number and most importantly the Coding index number are within the TestBook / T4 database, it will not go near attempting coding an X5 or other BMW ECU.
There are even examples within LR compatible ECU's where the coding index is different and must not be crossed, such as the V11 and V12 Light check modules.
In the Faultmate, if a system is interchangeable with a BMW one, such as the park aid module is, we do provide all the relevant coding data for that ECU for all known applications
EG
https://www.blackbox-solutions.com/shop/help/SM056.html
I hope this helps