When humans cry, the respiratory system will exhale very long and rarely inhale. Therefore, the longer they cry, the less the oxygen stock in the lungs (and body). Although it is inevitable that they will be mixed with intermittent inhalation, the oxygen stock in the lungs is much less than that in the exhaled lungs.
At the end of exhalation, a considerable amount of air (about 80-90%) remains in the alveoli in the normal lungs. Even if the inhalation is stopped for a short time, the circulating blood in the lungs can still bring the oxygen in the lungs into the blood, so there is no hypoxia. However, when crying violently, the air in the lungs is forced out of the body (crying is exhaling), and the remaining oxygen in the lungs is relatively small, so the amount of oxygen available to the blood will naturally decrease.
Moreover, intense crying is intense exercise, and the consumption of oxygen in the body will be much higher than that in calm, so the oxygen content in the blood is also low. When crying, the muscles of the chest wall and abdomen will squeeze the lungs, and the pressure in the lungs and pulmonary blood vessels will rise. Therefore, in the blood circulation, the blood flow in the lungs will be reduced and the oxygen uptake capacity of the blood will be reduced, so the oxygen content of the blood supplied to the brain and trunk will be sharply reduced. This is the main factor.
To sum up, people cry intermittently. Sometimes they don't have time to breathe, so they will choke.