How can I get rid of this "frog neck"? Not a fat issue (Photo)

Of course, an examination is necessary to gain more information. The following should not be considered a medical opinion.
A surgical procedure called a "corset platysmaplasty" is performed through an incision under the chin that hides in the natural creases of your neck.  The platysma is a muscle that runs from the cheek, over the neck, and down to the chest.  The muscle is attached by ligaments to the skin, and both sag simultaneously.  With a corset platysmaplasty, the surgeon can also look at the fat and digastric muscles below the platysma, to see if they are contributing to the neck sag.  The surgeon can also see if there is a need to release muscles from the hyoid bone. 
A chin implant would give your chin a slight forward position that would increase the length of your neckline and give you a more balanced appearance.

The only reliable way to get sharp neck definition is through surgery. Kybella would be unlikely to do this for you. See a qualified specialist to determine proper expectations for surgical results which depend upon your unique anatomy.

Hello and thank you for your question. Based on your
photograph, you may benefit from a chin implant which can be performed for
improved jaw line contour.  I always use a medpor implant with screw
fixation, which stays in place over time.  A silicone chin implant has a
high rate of malrotation over time.  Make
sure you specifically look at before and after pictures of real patients
who have had this surgery performed by your surgeon and not just a computer
animation system. The most important aspect is to find a
surgeon you are comfortable with. I recommend that you seek consultation with a
qualified board-certified plastic surgeon who can evaluate you in person.
Best wishes and good luck.
Richard G. Reish, M.D. FACS
Harvard-trained plastic surgeon

Surgery would be the most controlled means of removing any unwanted fat and redraping the skin of the neck. A chin implant can also influence and improve neck contour, depending on your goals. An in-person consultation with an experienced surgeon would be the best way to identify your concerns. 

 It's important to know that there are fatty deposits in the neck located above and below the platysma muscle. The fatty deposits located underneath the muscle must be removed with surgery. A platysma plasty can significantly improve the jawline. There also appears to be a week and recessive chin profile for which a chin implant can improve the projection of the chin which also gives better foundation for the soft tissues in the neck. For many examples, please see the link and the video below.