Yes! In fact, the privileges originally applied to current students as well.
The modern university system in the West is born in the 12th-13th century, out of the so-called "cathedral schools." Not at all coincidentally, this is the era of a massive revival of interest in written law, both civil and canon (secular and Church); and the bureaucratization of governments (which includes the Church). The basic goal of these early universities is to train preachers for the laity and clerks to staff bureaucracies.
Per medieval law, a "privilegium" is a private law, which means a law that applies only to one person or group as opposed to all of society. In 1158, Holy Roman Emperor Frederick Barbarossa stuffed a privilegium entitled Authentica Habita into an otherwise-unrelated council decree. Habita granted safe conduct to scholars traveling throughout the HRE for the purposes of study (which includes part of Italy during this time). It also granted scholars the right to be tried for alleged crimes by a judge of their own choice, rather than automatically subject to the civil jurisdiction of the town that accused them.
This was important for the scholars because the 12th century was rather much an era of petty violence, and traveling scholars might well find themselves kidnapped and held hostage by one lord or town to extract payment or leverage from another lord. It was important for Frederick because among his closest counselors were legal scholars from the Bologna schools--who had written effusively in support of his imperial claims.
Just as Frederick's legal scholars built on Roman law traditions when they shaped the scholarly privilegium of the Habita, other principalities in Europe would build on that decree to extend their own protections to scholars. This movement from secular law coincided neatly with an unshakeable principle of medieval higher education. Thanks to the roots of the university system in cathedral schools and ongoing ties between the universities and the Church, students (and by extension, teachers) were necessarily minor clerics. The privilegia offered to clergy were theirs, especially the right to be tried in an ecclesiastical rather than civic court for any crimes.
Well, concentrate a group of adolescents into one place and you're going to end up with shenanigans, right? From 1200 on, there is an escalating trail of sources detailing "town-gown" friction in the Middle Ages. Tavern brawls, riots, murders, landlords that make the Thenardiers look like the Holiday Inn.
Increasingly, legal scholars interpreted and reinterpreted Habita and its sister privilegia to cover more and more legal territory. Statutes proliferated. Civic authorities who arrested students in 1210 Paris, for example, had to make sure they were kept in a nice jail. In 1265 England, Oxford students were exempt from jury duty. 13th-14th century legal scholars tried very hard to blanket Europe with statutes asserting that scholars did not have to pay taxes. By 1500, burning down the residence of a scholar was considered not just arson but sacrilege, because of the holy books he was assumed to possess.
Why would Philip IV, desperate for funds to support his campaign in Flanders, free France's intellectual community from paying the war tax? Well, again, the people advising Philip down this path were scholars and university graduates themselves.
Additionally, though, privileges to universities and their corporate body (students and teachers) actually played a role in intranational politics. It was a chance for king/emperor and arch/diocesan Church (arch/bishop-level) to assert their authority over local government (town and parish).
Of course the high-minded explanation offered was the need for scholars to devote themselves to study. And while we can appreciate the more romantic tinges of this notion, there is a practical core: the faster students get through university, the more clerks are available to staff princely courts, the more preachers can get out in the field to fight heretics and save souls--increasing, respectively, civic and ecclesiastical authority and power.
The university degree, or rather degrees, themselves awarded additional privileges. Most importantly for the universities themselves, the ability to teach at various levels. However, it was a fairly regular occurrence in the Middle Ages to complete most of an education without paying the hefty additional fee to receive the degree. It was a question of whether the student needed the credential (for example, to continue education in one of the higher faculties of law, medicine, or theology) versus what he (always he in the Middle Ages) could afford.
From around 1500 or so, however, civic governments exercised increasing control of universities as the Church's iron control fluctuated. Local nobles sat in judgment of internal university disputes, as in Ingolstadt (in Bavaria). Growing attention to public order meant the excesses of students that earlier privilegia had protected needed to be brought under control. The rights, responsibilities, privileges, and immunities of the Middle Ages were gradually etched away.
As I write this from a university server: it's nice to dream, isn't it?
This is still relevant in some professions even today
See this discussion of the phrase by a recently graduated "Doctor of Medicine"
http://www.acpinternist.org/archives/2012/09/residents.htm
variations are common
all the rights and privileges thereunto appertaining
For the United States things are pretty limited, but generally the rights and privileges are going to be some sort of guarantee that the degree granting institution will verify that you were in fact awarded the degree in question. Since a lot of jobs now require degrees this is a non-trivial right. Additionally, alumni usually get some sort of perks such as library borrowing privileges, access to athletic facilities, etc. but that is very dependent upon the school.
see also this discussion of the phrase
http://www.unl.edu/gradstudies/current/commencement/200808
It is language that comes to us courtesy of British constitutional law and decrees and patents and grants made by the British crown.
It might be helpful to know where your degree came from (i.e. country and specific university, if you're willing to share). The MA (Oxon), for instance, describes an academic rank rather than any actual additional degree of study and entitled the bearer to, depending on the particular time period, either engage in teaching themselves or to be a full member in good standing of Oxford's university congregation: https://en.wikipedia.org/wiki/Master_of_Arts_(Oxbridge_and_Dublin)
So depending on where you are certain degrees or honors could certainly entail new rights, responsibilities or privileges.