I am confused about the usage of the word "type". Take the following for example:
- A maine coon is a type of cat.
which most people believe is correct standard English. But, dictionary definitions define "type" as a group with similar qualities. So, a "type" is really plural, not singular. This seems to make sentence 1 wrong, with "a maine coon" being singular. It seems that the correct version of sentence 1 would be:
- A maine coon is a member of a type of cat.
What do native speakers think? Is sentence 1 correct after all because it is idiomatic , and is sentence 2 unnatural?
Short Answer: Sentence 1 is correct, standard and idiomatic English. Sentence 2 is most unnatural.
Sentence 1 is so correct, standard and idiomatic that a sentence using its exact structure begins the description of the titular subject of the 2015 children's reader Life Story of a Salamander, published by the educational publisher Capstone.
And type is not "really plural," it is singular. Its plural is types. What appears to be throwing you off is the word group, which is notionally plural. Many dictionaries do not use the word group when defining type; instead they use such words as category, kind, sort, which—like type—are singular in number.
"dictionary definitions define "type" as a group with similar qualities" (my emphasis)
This is true of definitions found in some dictionaries. For instance, Cambridge Dictionary proffers such a definition when it defines type as
a particular group of people or things that share similar characteristics and form a smaller division of a larger set (Link)
and MacMillan offers the following:
a group of people or things with similar qualities or features that make them different from other groups (Link)
With these definitions in mind, it is natural that the sincere learner might be confused when coming across sentences such as
A maine coon is a type of cat
A salamander is a type of animal
TO WIT: If a type is a group and a group must be plural, how can a (which refers to one) maine coon or a salamander be plural? This almost sounds like a trick question.
It is worth noting, first, that although group is notionally plural (it refers to more than one item), it can be grammatically singular. Group is a collective noun and may take either a singular or plural noun, depending on what you mean, that is, whether you construe group as a single unit or as a disparate throng.
Next, Oxford online, noting that it is "treated as singular or plural," defines group as
A number of people or things that are located close together or are considered or classed together (my emphasis) (Link)
Note Oxford's use of classed together as part of its definition of group.
This brings us to group as a synonym for type, kind, sort, etc., and we can construct sentences similar to the originals as
A type or kind or sort or category or class or group of cat is a maine coon.
A type or kind or sort or category or class or group of animal is a salamander.
This is not the most common use of group; for instance I doubt we would often find
A maine coon is a group of cat
but it is just possible, if we use group as a synonym for type, kind, class, category, etc.
Definitions in other dictionaries avoid this source of potential confusion by not using group in their definition of type. Here are two found in the OneLook Dictionary search for type.
A category of people or things having common characteristics
a kind, class, or category, the constituents of which share similar characteristics (Collins)
Meanwhile, the Oxford English Dictionary gives
6a A kind, class, or order as distinguished by a particular character (OED)
The structure that you ask about is not only correct and standard and idiomatic, it is common and plentiful. Examples abound in, for instance, Google Book searches.
Note that some examples are straight from the world of linguistics:
(140) An X is a kind/type of Y
Thus a lion is a type of animal, a snake is a type of reptile, etc.
Many others, not surprisingly, are from the worlds of semantics, philosophy, cognitive science, etc that deal with definitions or representations of classes or types.
Compare: if I tell one of my grandsons that a lynx is a type of cat, he immediately knows a great deal about lynxes because he already knows a great deal about cats. (Link)
One comes from the extended usage note regarding the words sort, type, and kind that is linked to from the definition of type found in the same Cambridge dictionary with which I started this answer.
A fastener is a type of metal button which fits together to join clothes, for example a coat might have fasteners.
The list of usages is potentially endless because it is such a common usage/structure:
a sparrow is a type of bird; a wasp is a type of insect
A poodle is a type of dog in the same way that a great white is a type of shark.
an ampersand is a type of punctuation...
A meatus is a type of canal.
In fact, a gel is a type of sol in an intermediate physical phase
A lid on a glass jar is a type of screw...A see-saw is a type of lever...a screwdriver is a type of pulley...an axe is a type of wedge (Four uses in the space of eight test questions)
A so-called kegger is a type of house party at which the attendees drink beer that is pumped from a keg into plastic cups...A rave is a type of dance party that involves dancing to loud electronic music played by a DJ
“Fast chess is a type of chess game in which each side is given less time to make their moves than under the normal tournament time controls of 60 to 180 minutes per player”
In contrast, a taxis is a type of orientation in which the animal directs its body toward or away from the stimulus.
An observational study is a type of study in which individuals are observed or certain outcomes are measured and no attempt is made to affect the outcome.
Etc., ad infinitum.