Could it be that this sentence using "type":
1a. A cat belongs to a type of animal.
is less common than similar sentences using "class" or "category" like the following?
1b. A cat belongs to a class of animal.
1c. A cat belongs to a category of animal.
I just wrote a long answer about why type and member don't go together. That answer explains that the word type suggests a model or ideal of which instances are imperfect or slightly varying copies rather than a whole or collection that contains parts or members.
The primary sense of belong means standing in an appropriate relationship to something greater or more fundamental, as property belongs to a person, a person belongs to a family, a member belongs to a club, dishes belong in the kitchen, a talented violinist belongs in a good orchestra, etc. The meaning is very similar to the possessive or genitive case, which can sometimes express the same relationship: John's fishing rods, the club's newest member, etc.
Belong doesn't clash with type as strongly as member does, but it clashes, so indeed your example 1a is less common. Usually in English you use the possessive or genitive in the other direction with type: "the cat's type is Maine Coon"; "the cat is a type of animal".
Just as the etymology of type explained why it leads people today to think of an abstraction as a separate individual from its instances, the etymologies of class and category explain why these words lead people to think of an abstraction as a group or collection.
Class in Latin meant a division of the Roman people into higher and lower ranks, or various divisions of armies or people in military service. Over the centuries in English, the word has been extended into many senses, but they all preserve the idea of dividing a large collection into smaller collections arranged by similarity, as when classifying, or distinguishing the highest-ranking individuals as when we say that someone has "class" or a book is "a classic".
This makes class a natural fit with belong and member. It seems axiomatic to say that people belong with others of their same class. Since a class is a (sub-)collection, each individual is a part. Unlike the metaphor of type, it doesn't make sense to think of a class without members. A class is its members. Its members are the ones who belong together because of their similarity.
The word category has a somewhat complex history, originating from an Ancient Greek word that today we would translate into English as "predicate": something that you can say about something. For example, in "Maine Coons are large cats", "large cats" is the predicate, said about Maine Coons. Over time, in English, category has come to mean the collection of all the things that the same predicate is true of. For example, the category of "four wheeled-cars" is all cars that have four wheels. Choose any car from that category, and "that car has four wheels" will be true, because that is how the category is defined.
Again, since the word category makes you think of a collection of things that have something in common, it has a natural fit with belong and member.
The cat is a type of animal. [Here, "the" is a determiner meaning that we are using "cat" as the name of a type.]
Cat is a type of animal. [Omitting the determiner also means that we are using "cat" as the name of a type.]
The cat belongs to the class of all animals. [In other words, "all animals" is a class, of which "the cat" is a subclass.]
Cat belongs to the category animal. [In other words, animal is a category, of which cat is a subcategory.]
Purrcolator belongs to the class cat. [In other words, Purrcolator is a cat.]
I would say that "type" is the more common of the options. "Kind" may also be used. Note that typically it's
A cat is a type of animal
EDIT (expansion)
A type is not something you belong to, it's something you are. A talent agent might say "He's a Tom Cruise type".
Classes and categories (and pigeon holes) are something you're in.
