Thanks MDK. Based on your positive ID of it as a Samurai, I did some more digging and it turns out that you are correct but not entirely correct.
First off, I found a picture from the movie that shows the logo:
Remember, this sequence takes place in Gao, India (see below).
As I stated above, it looks like something that Suzuki would build but there were two reasons I didn't think it was a Samurai:
1. The dimensions are different from the U.S. Samurais that were everywhere during the late 80s.
2. The front logo is something I have never seen before. Every Suzi I have ever seen in the U.S. either just said "SUZUKI" on the grill (a la the FJ's "TOYOTA") or had the Suzuki "S" logo, like this one:
Turns out, the truck in The Bourne Supremacy is indeed a Samurai but it is a Samurai LWB (stands for Long WheelBase), a vehicle that was not sold in the U.S. but was sold in many other countries under a variety of different names.
The version sold in India was called the "Maruti Gypsy." Here is one with a softtop:
Lo and behold, this is the official Maruti logo:
A bit dirty/scuffed up, but that is the same logo as the movie truck.
Mystery solved, thanks for the help.