Not used by all processes without some save and restore as the context switches from task to task.
It appears you are starting a class on OS design. That topic will not fit in this small space (or time.) Give yourself a few years to get the deep understanding.
How to ensure data integrity in General Purpose Registers (GPR registers) given their limited number (for x86 their number is 6),and on the other hand that they are used simultaneously by all processes?