Yes, that looks like the original transmission.

If you look at the block in the top right-hand corner of the nameplate -

The '2' indicates 2002 model year

The 'G' means it's a GM 5L40-E transmission

'XE' is the code for the E53 3.0i transmission for a vehicle built between 10/01 & 10/02.

('TZ' for 08/99 to 10/01, 'XU' for 10/02 to 10/03 & 'YZ' for 10/03 to 09/06)

Phil