A friend of mine got Frozen Shoulder (Adhesive Capsulitis), and an incorrectly administered flu shot was suspected to be the cause. Apparently there are right places and wrong places to give the injection. Relevant article.
For what it’s worth, I’m an insulin dependent diabetic and have been injecting myself in the arm at least twice a day for 30 years now. Sometimes the needle hurts like a bee sting, sometimes I feel absolutely nothing at all, most of the time it’s somewhere in between. IOW I think it’s simply determined by the semi-irregular density of nerves you happen to hit in any particular injection site.