First and most importantly, pain of this sort should never accompany exercise, whether we are beginning or we are more advanced. If you are experiencing pain, there is something wrong with what you are doing.
What you are describing is symptomatic of costochondritis, an inflammation of the medial costal cartilages. Since the (sternocostal head of the) pectoralis major muscle originates at the sternum and costal cartilages, and since the pectoralis major is the prime mover involved in dips, acute or chronic injury due to overloading or overuse, respectively, will likely present itself at this site.
It should be noted, of course, that a diagnosis of your injury can only be made by a physiotherapist or sports doctor.
Despite your level of conditioning, dips present unique loading characteristics that—especially dependent on the way in which you are performing push-ups and bench press—may be quite distinct from what your body is accustomed to. Most notably, the line of pull in the dip is more vertical than horizontal, and elongation or pre-stretching of the pectoralis major is typically far greater than it is with forward presses.
The German hang, whilst appearing to be a more advanced form of dip, is really a distinct exercise. And as with the push-ups and bench press, the angles and orientation of the humerus is very different. Specifically, at the extreme of humeral hyperextension—that is, at the bottom of the movement—the humerus is internally rotated. Since the pectoralis major inserts at the anterior of the humerus in the bicipital groove, internal rotation reduces tension on the pectoralis major significantly, transferring much of that load to the passive structures of the anterior deltiod and glenohumeral joint capsule. By contrast, in the dip, the humerus is fixed in a neutral or externally rotated position, forcing the pectoralis major to bear most of the load and tension.
I hope that explanation is helpful.