A "school" of magic is just a category of magic; an academy which teaches multiple schools of magic is valid and appears in D&D canon.
A "school" of magic is not an actual institute of education, but rather the term refers to a type of magic. A place of magical learning is typically called a magical university, college of wizardry, or some other name.
This is best described in the AD&D 2nd edition Player's Handbook (revised, p.44):
Spells are divided into nine different categories, or schools, according to the types of magical energy they utilize. Each school has its own special methods and practices.
Although they are called schools, schools of magic are not organized places where a person goes to study. The word "school" identifies a magical discipline. A school is an approach to magic and spellcasting that emphasizes a particular sort of spell. Practicioners of a school of magic may set up a magical university to teach their methods to beginners, but this is not necessary.
Colleges or guilds may specialize in one school of magic, although a Hogwarts-style academy which teaches all schools of magic in separate classes is also valid. Colleges of both types appear in D&D lore.
For example, the Bleak Academy (Tome and Blood p.25) specifically teaches necromancy, while the Arcane Order (Tome and Blood p.23; and detailed at length in the AD&D 2nd edition book College of Wizardry) teaches all schools of magic. This is specifically stated in College of Wizardry, p. 25-26:
The wizards of the Arcane Order of Enchantment & Exposition are a mixed group. Not every wizard specializes in the same school of magic. In fact, of each of the nine schools of magic described in the Player's Handbook has its adherents in the Arcane Order ...
Of course, some decide that the walls of specialty are too constricting. These wizards become mages; that is, general practicioners of magic. In this case, the mages are taught by a variety of different tutors and advisors, never sinking too deeply into any one school's philosophy of magic.
For completeness, other editions of D&D largely concur with AD&D's definition of a spell school. D&D 3.5 Player's Handbook, p. 172:
A school of magic is a group of related spells that work in similar ways.
D&D 5th edition describes each school of magic as a separate ancient arcane tradition (Player's Handbook, p.115):
In some places, these traditions are literally schools: a wizard might study at the School of Illusion while another studies across town at the School of Enchantment. In other institutions, the schools are more like academic departments, with rival faculties competing for students and funding. Even wizards who train apprentices in the solitude of their own towers use the division of magic into schools as a learning device, since the spells of each school require mastery of different techniques.
D&D 4th edition uniquely broke from this tradition, and its Player's Handbook only uses "school" to refer to an arcane academy:
Alternatively, you might found a school of magic, for which you serve as a rarely seen headmaster.
AD&D 1st edition's Players Handbook didn't yet use the term "school", instead referring to it as "type" of magic. In the earlier question What is the origin and meaning of D&D Beyond's spell school symbols?, we see that AD&D 2nd edition introduced both the term "schools" and the symbols for those schools. This may be because 1st edition only include one specialist wizard, the Illusionist, while 2nd edition had the possibility of a specialist in any type of magic, and thus there needed to be a better name for types of magic, and the concept of opposed schools represented in the diagram.