If a child is very young, I think that if you can get the child to understand that the four legged creature is "dog", that is more important than getting the very young child to understand the sentence structure and the difference between the creature's type or the creature's name. Then, after you see eight more creatures named "dog", your child will intuitively start to understand that the word refers to that type of animal, and not just one single animal.
Normally, I think of the name of an animal as an identifier of an individual creature. e.g., Spot, Rover, Daisy, Snoopy.
However, people also refer to the "type" of an animal (e.g. "dog") as the "name" of the animal. (In this case, I would say the word name is referring to the name of the type of the animal.)
e.g., the super-hilarious "Name an animal with three letters in its name" [video on YouTube]. This could also have been said as, "Tell me about an animal where the name of the animal's type is three letters long." (Except, this native speaker did want to complete the sentence quickly.)
This is a tricky question, because the article is definitely required to identify a type of animal. e.g., "That is a dog" is the correct way to say the type of dog. If you don't use the article, then it sounds like a name. e.g., "That is Scoopy-Doo" is the correct way to say the name. However, saying "That is dog" sounds acceptable (not wonderful, not great, but rather acceptable) because people do speak like that, sometimes.
Leaving off the article is not preferred, though. If the article is left off, then that is less clear.
"Daddy! What does 'a schizophrenic' mean?"
In general, I think it is more clear to use the word "schizophrenic" as an adjective. For example, "a schizophrenic person". So, "That person is schizophrenic" makes sense, because the form of that sentence is similar saying "That person is tall". However, I do believe the word "schizophrenic" can also be used as a noun. So, in the phrase "a schizophrenic [person]", the word person is optional. So, saying "That person is a schizophrenic" could work.
Those sentences are both valid because the word "schizophrenic" can be used in both ways. Some other words are not like that. For example, "He is professional baseball player" is bad. That needs the article "a". Saying "He is a slow" is also incomplete. ("He is slow" is fine. Or, "he is a slow person".)
- "He is an African". Okay, African is a noun.
- "He is African". Okay, African is an adjective.
- "He is an American". Okay, American is a noun.
- "He is American". Okay, American is an adjective.
- "He is a European". Okay, European is a noun. (You get two extra points if you realize why I said "a" instead of "an", despite the next word starting with the letter E.)
- "He is European". Okay, European is an adjective.
- "He is French". Okay, French is an adjective.
- "He is a Frenchman". Okay, "Frenchmen is a noun."
- "He is a French" -- bad. A person is not a French. The word "French" is not a valid noun.
In summary, deciding to use the article (the word "a") affects the structure of the sentence, by deciding whether the next word is a noun or an adjective. Sometimes you can get away with using it or not using it. This is not always true.