How does the shoulder joint stay in place?
Because it's

held together by muscles, tendons, and ligaments.


(image from Wikipedia.)
It's somewhat complicated how all these parts contribute to holding the joint in place because it depends on the position of the joint. E.g.

Shoulder muscle forces are usually powerful stabilizers of the glenohumeral joint, especially in mid-range positions when the passive stabilizers are lax. However, muscle forces can contribute to instability as well. Certain muscle forces decrease glenohumeral joint stability in end-range positions. We found this to be the case with both active and passive pectoralis major forces.

That study has some data based on computational models and actual tests on cadavers.