The caulk stuff that is black now is probably butyl rubber tape. It is like a sticky putty. When new it is grey but the exposed edge turns black from dirt & age.
To use it you would have to remove the vent frame and clean off the old stuff. Then you form a flat gasket with the butyl tape and reinstall the vent frame. Use something like a plastic putty knife to trim off any excess butyl tape that gets squeezed out after all the frame screws have been tightened.
Caulk can be hard to remove should you ever have to. An RV shop would be able to recommend a caulk that adheres to paint and bare metal and is not affect by the long term exposure to the sun. I'd go with the butyl rubber tape method myself.
The butyl rubber tape is wide so it might still be OK under the frame.
I can't see your lower vent that well in the photo.
Mine has a flexible clear plastic cover over the hinge. Water can't get under it. I wonder if you could do something like that? Or maybe closed cell weather strip that would compress and seal between the door and the frame. You would need to get the right thickness.