Just as a note: medicine and law are not separate degrees everywhere. Where I come from (Brazil) , law and general practice medicine are just undergraduate degrees like engineering, physics or history. Specialized branches of medicine are separate degrees though, acquired during residency.
The first European universities appeared during the 11-12 century and in the beginning things were very simple. The people who achieved university education then were mostly known as "scholars". A scholar could afterwards stay with the university to lecture.
With time, the universities grew and various faculties began to appear. Each faculty provided studies in a particular field, much the same way as a particular guild would teach a craft. Just as in the guilds, the person who reached the top level of university education would achieve the status of a "master".
At first there was only one degree, known as Master, or Professor, which simply meant this was the "teacher", the person with most knowledge and the one in charge. For example, in the 12th century, the oldest European university, in the Italian town of Bologna, awarded degrees such as Doctor of Civil Law. They also had simolar degrees in Cannon Law, Medicine, Grammar, Logic and Philosophy.
In the 14th century, the Bachelor was a title already given to people in training elsewhere, such as a junior member of a guild, so at the universities also the student-teacher started to be called Bachelor. The Bachelors were more than regular students but not quite as knowledgeable as a Master. They would have studied for 3-4 years and successfully passed exams held by masters, but had not yet reached the final stage of academic education. Only after final exams held by the chancellors board, they could receive Master's or Doctor's degree and a certificate allowing them to teach elsewhere.
Since the universities were quite independent, the degree names could differ. Sometimes, the faculties gave different names to the highest degree they would award. The top degree in Medicine could be a Doctor, but the top degree in Theology could be a Master, both being equall in academic standing. Or, a Master of Arts in France would have gone through about the same academic studies as a Doctor of Philosophy in Germany. The doctorate degree was first granted at the university in Paris in 12th century, but only came to America in the 19th century, first in Yale and then in Harvard.
Further differences between Master and Doctor appeared due to the change in academic requirements. In England, Master of Arts became important in order to proceed to other studies, thus making the Doctor a title to be achieved later, as the most advanced level. A Doctor's degree was more focused on research in one's chosen subject and led to becoming an expert in the field.
Further reading, "A History of the University in Europe" by Walter Ruegg
(Hi. First time answerer on /r/askhistorians. I have a BA in Soviet History but have since made a study of modern Chinese history. Hope this answer is ok.)
This may not be the answer you are looking for (and I mention it for interest's sake) but something analogous to our modern degree system was practiced in imperial China for many centuries.
Imperial China was in many ways, and at many periods in history, highly meritocratic. A system of imperial examinations meant that any literate Chinese man could demonstrate their learning and, depending on the level achieved, earn a place in the civil service.
The examination system dates back in some form as early as the Han dynasty (206 BC – 220 AD) and further back into mythology but came to its modern form in the Tang dynasty (618–907 AD). After variations in practice and a hiatus brought on by the fall of the Ming dynasty, the Manchu emperor Shunzhi reinstated the system in 1646 and it was practiced in some form right up to the end of the Qing dynasty and the founding of the Republic of China in 1912. The exams were meritocratic in the sense that broadly, anybody could sit one, and the results were assessed after a rigorous process of anonymisation, so that neither bribery nor nepotism could favour a candidate.
Examinations were generally held every three years. There were several levels of degrees and there are parallels to what we are used to today:
As a side note, the exams were based on rote study of the classics and had a highly moral slant, and were in many ways highly stylised and would hardly compare to modern ideas of good scholarship. Neither were they highly specialised like a modern degree.
I found a paper, cited below, that suggests that written examinations were not practiced in the West until 1702 (Cambridge), and that there is some evidence that knowledge of the Chinese system had reached the West and may have influenced western examination systems, for instance the British civil service in India.
Sources: Jonathan Spence, The Search for Modern China (1991)
Teng S. Chinese Influence on The Western Examination System, Harvard Journal of Asiatic Studies 1943:267.