Reading about Al-Azhar University, whose page says it dates back to the tenth century, making it older than Bologna. Why is it not counted as a university?
Because many of those universities, including European universities, are quite dissimilar from what we understand to be a university. University and center of learning are not synonymous, and the University of Bologna is the first university in the modern sense of the word, and was the institution to coin the term.
Al-Azhar was originally built as a mosque. Fatimid Caliphs encouraged scholars to study in the mosque, and soon these scholars began to teach classes. On the other hand a university is a guild/corporation-like organization that has a unified curriculum or objective in its classes. There's a group of people who decide "Okay, this is how this institution will be run, we'll have this guy teach astronomy, this guy teach rhetoric, we'll charge this amount of money etc." While Al-Azhar has converted over to a secular university in the modern sense in the last century, the University of Bologna was founded as such, making it the oldest. It was also autonomous, and awarded degrees. Otherwise, we would call the schools of Socrates and other ancient/medieval schools.
Because "university" is not synonymous with institution of higher education. While there are many different types of higher learning institution and several predate the university, the university has its origins in the High Middle Ages.
As has been said before, a university is not the same as a school of higher learning. Because nobody stated explicitly what qualities designate a university, I will give a short definition:
I quote Walter Ruegg, wo paraphrases a university as "[...] a community of teachers and taught, accorded certain rights, such as administrative autonomy and the determination and realization of curricula (courses of study) and of the objectives of research as well as the adward of publicly recognized degrees" [volume I, page xix].
Institutes in the Islamic world did e.g. not adward degrees, hence they can not be considered a university. This doesn't devalue the quality of their teaching though! Calling an institution "university" is more a question of nomenclature and terminology.
Sources: The four volume History of the University, Hilde de Ridder-Symoens and Walter Rüegg (eds.), Cambridge University Press, between 1992 and 2011
hi! you'll find a bit more on in the FAQ section Life at University:
It's because the University of Bologna IS the oldest 'university'. The term university means, "community of teachers and scholar" which is derived from the latin, "universitas magistrorum et scholarium", but this isn't an etymology question so it doesn't matter.
University != Institute of higher learning.
The university emerged in the high medieval ages period (1001–1300) and has roots in the christian monastery and cathedral schools which had existed for half a millennia (early 6th century) before the Bologna University was even founded.
Therefore, other institutions of higher learning such as Plato's Academy (387 B.C.) and others from the Greek/Persian/Roman/Byzantine/Indian/Chinese/Islamic world are not included in this list due to their "cultural, historical, structural and juristic dissimilarities from the medieval European university from which the modern university evolved."
Here's a nice article you might enjoy on higher institutes of learning in the ancient world:
http://en.m.wikipedia.org/wiki/Ancient_higher-learning_institutions