All pressing movements involve flexion and horizontal flexion/adduction of the shoulder, protraction of the shoulder girdle, and extension of the elbow. It is commonly assumed, therefore, that all variations of presses load the shoulder flexors (pectoralis major and anterior deltoid), shoulder girdle protractors (pectoralis minor and serratus anterior), and elbow extensors (triceps brachii). And this is mostly true, but with one caveat: it is only true of presses with a closed kinematic chain.
Elbow extension in open-chain exercises like the dumbbell press can be entirely passive, in such cases that the elbow lies beneath the load—or more precisely, when the load force vector passes through the elbow. That is, the angle of the elbow is not forced, but placed. (As an analogy, imagine balancing a broomstick on your finger. There is no connection, no mechanical control—just positioning of the balance point relative to its centre of mass.) In the most extreme examples, the dumbbell is controlled entirely from the shoulder. And a good way of developing this control is to exchange the dumbbells with shots balanced in open hands. If there is any active involvement of the elbow extensors, the shots will roll out of the hands!
All of that said, however, both the elbow flexors (biceps brachii, brachialis, and brachioradialis) and extensors are involved in controlling and stabilising the position of the dumbbell, with greater instability demanding greater activation of those muscles. Many lifters further habitually position the dumbbell outside of the line of elbow, thereby actively loading either the elbow flexors (lateral over-balance) or extensors (medial over-balance).
If the dumbbell is over-balanced laterally, a component of the load will unavoidably be borne by the three elbow flexors, with the single-joint brachialis and brachioradialis controlling elbow flexion and the multi-joint biceps brachii contributing to both elbow and shoulder flexion. Due to its origin at the coracoid process, the short head, particularly, is susceptible to loading as the humerus is extended horizontally.
So the pain you are experiencing is likely the consequence of overloading or overworking the biceps brachii—and most likely the short head due to its point of origin and its role in shoulder flexion. And the reason that the pain may not be presenting during bicep training is that in the traditional ‘biceps’ exercises, the other unaffected elbow flexors are able to assist and dominate. The pain might therefore be avoided with modification to your technique (balancing the dumbbells more medially) and, of course, rest and recovery.
Please understand, of course, that a diagnosis can only be made by a qualified sports physician, but I hope that this answer gives you some insight into what might be happening.