first things first - what sort of feedback is on the machine? does it use linear glass scales or rotary encoders on ballscrews?
when it is referencing - does it travel always to some/same point in space, a switch, or just incremental movement from where you left it (less than 100mm)?
I'm asking because TNC can be configured in variety of ways, some will behave sort of like you're describing, and will not work correctly when machine parameters are lost and copied from a similar machine
do a simple test, after turning on both machines, allow them to reference, once in the manual mode, press [MOD], use up/down arrow buttons to get to the "display position" setting (I don't recall exactly what the menu item name is, but should be self explanatory), press [ENT] and it will cycle between (if I remember correct) NOM/LAG/ACT/ABS, switch it to ABS and compare what the X/Y/Z coordinates are displayed on both machines - I'll guess and say they are quite different between the machines
regarding the quote - I'm not that familiar with all versions of the 355, but there is a parameter which you can find if you download the service manual, that makes the TNC run the PLC program from either RAM or eprom, default should be from the eprom, RAM is used, as I tried to explain earlier - for debugging, once the program runs reliably, it is downloaded from the control, sent to Heidenhain, and they compile it and burn an eprom for you for a price - hence this program is written in such a way for it to be universal between same model of the machine, and then you adjust machine parameters for each individual machine
I'll post later (if I'll find my notes) how to do this, but it involves some code that allows you to first load the PLC program from the eprom into RAM, you can then view and edit it (in RAM, eprom obviously stays as it is)