The only data on a hard drive in the PS2 that gets keyed to a specific PS2 is any program that is bootable from the HDD from the Browser without a disc. HDLoader and programs it installs do not fit that description. I move my drive between my US and Japanese systems for various installations, and also use friends drives to install imports for them. The only odd behaviour is that the license agreement appears when you boot HDLoader with a hard drive that was last booted on a different system, but you don't need to reformat at all and you game list will be intact and fully functional.
My PlayStation2 HDD guide.
NTSC J SCPH 30000 with HDD and PSBBN 0.32
NTSC U/C SCPH 30001R Astral Blue LE PS2
Final Fantasy XI status:
Server - Ragnarok, Race - Mithra, Jobs - BLM 72 WHM 51 SCH 39 SMN 25, Nationality - Windurst, Name - Bungiefan