Many do not realize that there is really no such thing in English as an uncountable noun. What English does is to use countable nouns in uncountable ways--and we then refer to those nouns as "uncountable" within that context.
Some examples are shown in the table.
|How many apples do you have?
||Doctor: How much apple did the baby eat?
|There are many waters which merge in the sea: ranging from rainwater to rivers.
||We should drink water every day.
|Bakers make many breads: Wheat bread, rye bread, and cornbread are just a few.
||Asians often think Americans eat bread in place of rice.
|The ways of the world lead nowhere.
||The way of life is the search of a lifetime.
|The learnings from the mission will help NASA plan for a future mission to Mars.
||Students' learning in school should be balanced.
|Types of flies include mosquitoes, gnats, midges, and house flies.
||She is not my type.
|How many additives are in processed foods?
||How much additive is necessary to preserve the food?
As these examples show, virtually any noun in English can be used within a countable or an uncountable context, including the word "type." As a countable noun, it can be plural in form.
Note that not all countable nouns will appear plural. I have used plural forms in the table merely for clarity. But "one apple" is just as "countable" as are "two apples."
One additional note: Grammarians will say that a list of items should all be of the same form. If one uses plural, countable nouns in the list, the entire list should be of plural nouns; but if one uses nouns in a singular or uncountable form, all nouns in the list should match this.
I like apples, mangoes, and bananas. (Correct)
I like mango, strawberry, and apple. (Correct)
I like pear, papayas, and pineapple. (Incorrect)
So to answer the question:
Can we use countable nouns to refer to "types" of things?
Yes, we most certainly can use countable nouns to refer to types of things.