In my opinion: no.

Theoretically, the capsule can be injured with the extreme ROM and excessive force on the glenohumeral joint during the same mechanism as (separate) impaction injury to the coracoid. Obviously, the coracoid itself is extra-capsular.

What do you mean by “hitting the processus coracoideus”? Do you mean hitting the front of the shoulder with the bar when it is coming down? The capsule is protected under the processus coracoideus, which itself is under the deltoideus and subcutaneous tissue. Furthermore, due to its sheet-like structure and orientation, it is only likely to be injured by penetrating trauma or overstretching (either acute traumatic or chronic attenuation). So no, it is not likely to be injured in the setting you describe.

During the chest press I feel something there. The rotator cuff I train so that would be hopefully in balance. So something else must be hurting there, so I was thinking about the capsule of the shoulder joint.

Pain during bench press frequently originates from the acromioclavicular joint. However, something else may be going on, and without a physical examination it is hard to give a precise diagnosis.