The pictures were taken after I attempted to clean with a diluted APC.

I have also applied Shin-Etsu grease.

Any thoughts on what it is . . . it is on both the driver's and passenger's side door jambs . . . and how to get rid of it?

Thanks

 

It's hard to tell in the pictures, but it appears to be dried on wax.

From the Meguiars website.

The best way to remove wax from vinyl or rubber is to use Meguiar's #39 Heavy Duty Vinyl Cleaner. #39 will emulsify and breakdown the dry wax residue. If you have a tough time locating #39, any all-purpose cleaner should also do the job. Agitating the cleaner by gently scrubbing the surface with a soft bristle brush, like a toothbrush, can help break up the wax

Once you have cleaned the vinyl and/or rubber surface, bring back the natural color and help protect it with an application of a vinyl/rubber conditioner and protectant like NXT Tech ProtectantTM, Natural Shine, Gold Class Vinyl/ Rubber Protectant, M-40 Vinyl and Rubber Cleaner/Conditioner, or Gold Class Trim Detailer.

Another cheaper alternative is a soft toothbrush and some peanut butter, don't laugh it works! Work the peanut butter using the brush into the trim, the oil will emulsify the wax and you can wipe the residue off. Follow up with a protectant of your choice.

 

You live in New Jersey...I wonder if it's a stain from the salt they put on the roads in the wintertime. If the weatherstripping in that area is a little worn, the salt could be sitting in the pores of the rubber.

Regardless, VP's suggestions are good ones. I'd try them first. What APC did you use? You could also bump up the dilution with less water, and see if that gets you anywhere.