I've seen people injure themselves doing a handstand, because they skip the part where they learn to bail out when falling forwards. Learn to pirouette or roll before you try free standing handstands, folks!
But if you do it safely, risk of injury is very low. Less than spraining an ankle running on wet grass (which football and soccer players do all the time).
It’s the finger, wrist and elbow joints and ligaments (your fingers need to act as toes, your arms are now legs) and not learning how to bail safely (if you flip over and don’t have the reflex to cart wheel out).
There's a reason why handstand pushups (often done against a wall for muscle-building/strength purposes) are considered an advanced shoulder exercise. It's sort of the calisthenics equivalent of the overhead press.
I didn’t say they didn’t. But that won’t be the main source of injury. There are smaller and weaker muscles in the chain if you want to achieve a free standing handstand and not just lean on a wall inverted.
Do free floating handstands for time and see what hurts more.