I annotated your photo of the cooler. This looks very much like a transmission oil cooler, which I have installed many of in SoCal - and have one on our 73 Mach 1 with its swapped in AOD. They are a good thing to have. The annotation on the attached photo has a "speech bubble" suggesting you follow the one oil line to see where it leads. There are actually two oil lines (the other is beneath the one I pointed to), and I am betting you find they both go to the transmission. One is the hot oil supply line (usually the top cooler line is the inlet), the other is the cooler oil return line.
I can't tell from the photo if the rubber lines leading from the oil cooler to their respective steel lines are made for hydraulic pressures, like we find with automatic transmission cooling systems, or is someone only used fuel line class rubber hose. If the rubber lines leading to/from the oil cooler are stiff or hard, and/or show any cracks, I suggest you replace the two rubber lines with automatic transmission rubber oil lines, which are thicker than fuel lines and can handle higher pressures than rubber fuel lines are designed to handle. I would also use new retaining hose clamps, as you do not need or want to dump your tranny fluid under any conditions.
This is also a great time to pull the auto transmission dipstick and see of the fluid is clear (likely red in color) and not smelling burnt, as opposed to looking brown or dark, and smelling burnt. If the fluid smells burnt and is dark (not red) you may be able to save the transmission from eventual damage by draining the tranny and torque convertor automatic transmission fluid (ATF), replacing the tranny filter, adjusting the band(s), and filling the tranny with fresh ATF (Type F, not Dexron). Do not over fill the transmission. I am not sure wat tranny you have, it depends largely on your engine size. Keeping your tranny fluid clean will help prolong its useful life. And the oil cooler is a really good way to help keep the ATF from overheating and in good condition. If you do not have an inch-pound (aka pound-inch) torque wrench you will want or need to have the band(s) adjusted by a transmission specialist. It is not hard to do, just a little tedious and requires an inch-pound torque wrench. If the tranny is not slipping you are likely find driving the vehicle to a tranny shop for them to adjust the band(s), but please do not neglect getting the band adjust(s) done, especially after putting fresh oil and filter in the tranny.
There is not a drain plug in the transmission oil pan, unless someone added it after it left the factory. Dropping the oil pan without slopping ATF all over the place with no drain plug is an acquired art form. Before reinstalling the oil pan with a new gasket be sure the old gasket is completely removed, and using a ball peen hammer flatten out the oil pan where the attachment bolt hoes are located.
For example of how messy the oil pan removal can be see the following YouTube video at 01:22 mm:ss:
https://www.youtube.com/watch?v=7pA-JqkF81E
Draining the Torque Converter takes a little extra work, and if you are not mechanically inclined, it may be best to have a shop change the ATF (all of it, to include the Torque Convertor) and filter for you.
Not to try to make anyone feel badly, but below is a cool link showing a 10 year old kid changing his C4 auto tranny fluid and filter, to include draining the torque convertor. He has developed a very nice way of carefully draining the ATF by lowering his oil pan very carefully. It is worth the while to see how he does it without losing control of the fluid as it drains out in a very controlled manner, starting at 00:16 mm:ss, and the fluid draining from the oil pan begins around 04:05 mm:ss. He then begins prepping for draining the Torque Converter at 19:40, by removing the inspection cover (he calls it the Torque Converter Plate), then
https://www.youtube.com/watch?v=7pA-JqkF81E
Later in the video he drains the Torque Convertor, installs the new filter, buttons up the oil pan, and refills the tranny with fluid. He has clearly learned a lot from his dad, and is on his way to learning a highly skilled profession. We need more kids like him, of both genders...
