In Hogwarts School of Witchcraft and Wizardry almost all students have their own pets like owls, cats, toads, etc. I understand that students need owls for communication, but is it necessary that all the students must have a pet of their own?
The exact wording in the Hogwarts letter is
Note “may”, not “must”. For example, Hermione doesn’t have a pet until the third year.
Most students also won’t have their own owl. The school owlery has a set of owls that anybody can use for post, or there’s the post office in Hogsmeade which is accessible to the older students.
The students aren't required to bring their own pets, its a luxury. The pets of Hogwarts function mostly as animal companions and sources of comfort to students who are spending a large portion of the year seperated from their families. Cats, toads and owls are the traditional pets, but some students got away with bringing other pets, as long as they weren't big or dangerous.
Dumbledore was exceptionally unique in his bond with Fawkes was a rare sort of bond.
Ron got away with bringing Scabbers because the Weasley line was well know for having "more children than they could feed" and being very poor. A rat was probably one of the cheapest pets that can be bought in the wizarding world. Scabbers was relatively small and harmless. I'm sure he'd have preferred a new owl over Hermes, who was very old and losing his vision so was always crashing into things, but his family couldn't afford a new owl. He was awestruck when Sirius Black purchased a new owl for him after the events in Prisoner of Azkaban.
Crookshanks wasn't a pure 'cat' as he was a hybrid of cat/kneazle.
Hagrid's 'personal' pet was a large black dog, which is excluding every other animal he though of as 'his' that were cared for on the Hogwarts grounds.
Most people forget Ginny's pet was a pygmy puff. Ron also owned the pygmy puff's predecessor the puffskein briefly, until it supposedly died after being mistaken for a ball and used during bludger practice.
I'm sure depending on the pet in question they'd be allowed outside of the traditional three as long as there weren't safety hazards relating to it. Like pet tarantulas, weasels, some other type of small to medium bird species, herbology prodigies may prefer potted carnivorous plants over mammals, or even keep a gold fish. I highly doubt anyone would think it was unusual if a slytherin favored a serpent pet - much like the Slytherin founder - as long as it wasn't aggressive or poisonous and didn't grown large during the student's time at Hogwarts. What is mostly taken into consideration is practicality, maintenance, and portability.
All in all, giving how some students skipped years before getting pet, its easy to determine pets were just a luxury and not mandatory.