In most fields in the US, you are admitted to a "program", essentially to a department. The decision about an advisor comes later. If you enter with a BA/BS then it can be much later (years). The first task is to take whatever courses are necessary to get you through prelim/qualifying examinations, which are all on advanced topics. Even some MS entries may need some courses and need to pass quals. About that time (bit earlier or later) you make arrangements with an advisor for your dissertation work. 

In some scientific fields the PI decision may need to come earlier, but, perhaps, only in general - with a group or lab, rather than an individual. 

But, yes, they are usually separate decisions. In some places in Europe it is quite different and "admission" to a program is the same thing as being "hired" by a PI. It isn't like that in US. 

Until you pass quals in US, all agreements with PIs/advisors are tentative and conditional.