As I previously wrote, I don't know if any law that prohibits you from buying the car using your passport.
If your relative uses their DNI to buy a vehicle, I believe it will have to be registered in their name. If that's the case, as the legal owner, they will have to get the insurance and if the vehicle exceeds a certain value, they will have to declare (to AFIP) the source of the funds they used to make the purchase. I believe they will also bear the ultimate liability for the actions of any other drivers as well as responsibility for any infractions, including moving violations like speeding.
If you are providing the funds, they can probably declare tha you are thee source, but that doesn't seems as transparent (at least to me) as you buying the car in your own name, using your own funds, unless you want your relatives to keep the car when you leave Argentina (and avoid additional cost and paperwork at that point).
In either case, the person whose name the car is registered will get a green card which, as far as I know, does not expire. Anyone else driving the car will need a blue card. I think they have to be renewed, but I'm not sure of the frequency or if a tourist without an Argentine driver's licence can get one.
Keep in mind that if you buy a vehicle in your name (and perhaps in the name of your relative) that you may only have 90 days to "legally" drive in Argentina with a tourist permit. As you indicate you plan to stay in Argentina for two years, is it safe to assume that you will be applying for temporary residency? I ask because temporary residency is required to get an Argentine driver's license and, perhaps most importantly, having an Argentine driver's license may be essential for insurance coverage to be valid after you've been in the country for more than 90 days.
The last thing you want to do is to give the insurance company a reason do deny a claim on a technicality.
