Why do they call it the bar exam?

Because when the day is done the judges and the lawyers all meet and drink together at the saloon across from the courthouse.  

Its just a phrase, to mean the judges bench in a court room.

The bar is the physical divide in a traditional courtroom which separates the participants in any trial from the public viewing area. 

Lawyers (or in many countries, “barristers”) traditionally present their cases from

positions just within the bar. 

So they work at the bar, serve at the bar, are called to the bar, pass bar examinations, and as already mentioned some are called barristers. 

For the record, a barrister is a specially qualified lawyer who only works at the bar presenting defence or prosecution cases brought to them by other lawyers and solicitors who are not bar-qualified. The documents they get given contain all the relevant facts necessary to brief them on the case they are presenting. That packet of documents is called a “brief” and is usually a key-summary of all of the documents relating to the case.

It also just happens that in pubs and other places where alcohol is served, a physical divider was long ago deemed necessary to stop the public from getting access to the alcohol to serve themselves without making payment. That physical divider is also a bar as it bars access and gives rise to the word barrier.

In nautical terms a physical divider impeding access to a port, harbour or a beach is also called a bar. Often a sandbar or a shingle bar.