Most organizations do say that they look at the behavior rather than the person when it comes to the type of punishment meted out. This view allows a more dispassionate regulated response to the same behavior no matter who is involved. It is like saying that no matter who the person is, the same punishment will be given for the same behavior. I believe that this is the fairest way to manage the people in an organization because favorites who receive a lighter punishment for the same behavior lead to disgruntled, angry employees.
Now, I do believe that, while management may say that they believe the view that the same punishment should be given for the same behavior, practice does not always follow that stated belief. Maybe there are extenuating circumstances such as missing too much work because the person has a chronic health condition. Maybe the current people have been trying to put their lives back together, and management believes that they deserve another chance. My belief in the current workplace is that management does not follow the stated view, but does whatever it chooses to do especially if the offender is part of management. Too many instances of favoritism or looking the other way are part of the public record.