I am puzzled because I am use to the transmission being a separate

That sounds like the transmission cooler that is built into the radiator has failed. If you look at the bottom of the radiator you will see the two cooler lines that run the transmission fluid through the system. When the cooler fails internally you get transmission fluid into the cooling system. To repair it you will want to do a couple things. First thing is to remove the two lines from the cooler in the radiator, then plug the fittings on the radiator, now use a section of rubber hose and some clamps to connect the two lines. Now refill the transmission and take it for a test drive to make sure the transmission is still okay. If that tests out good you can do the actual repair. For that you will need a new radiator and some new coolant. However you will need to flush the cooling system first to get any of the transmission oil out of the system. To do that you can use the old radiator. Drain out the coolant that is in it, then fill it with water with a small amount of Dawn dish liquid in it. Let that run through the system until the engine gets warm, Shut off the engine and let it cool down. Drain out the water/dawn mix, now add clean water and repeat until you get out only clean water. Once you get all clean water you can remove and replace the radiator and connect the transmission cooler back up. Then add in new coolant and top up the transmission.
While the engine and transmission are two pieces many cars with automatic transmissions use a cooler like this as well.
Another option would be to install an external cooler and plug the connections on the radiator but you still need to flush the cooling system to get the oils out. They can damage the hoses in the cooling and heating system so you need to flush them out.

That car uses a few different radiator designs, one for a single cooling fan, one with two and those are both available for the manual or automatic transmissions. Replacement of both is similar. Drain the coolant or water out, remove the shroud over the radiator, then remove the mounting bolts from the brackets. Next you need to remove the screws that secure the AC condenser to the radiator. Once that is done you disconnect the hoses from the radiator. And can lift it out of the rubber mounts at the bottom. Reverse the process to put the new one in. Now because you had oil in the system I would replace both the upper an lower hoses and possibly the heater hoses, but if this was something that just happened and you caught it quick they may be OK. You will likely need to replace the transmission cooler lines or at least cut the ends off where they were attached to the original cooler fittings as they can get damaged by the tubing when removed.