1. "A" narrator is not necessarily "the" narrator.
A narrator is someone who tells a story. This can be anyone. If my grandfather tells me a story of his experiences during World War II, then he is narrating a story, and therefore a narrator.
This is based on nothing more than the semantical definition of the word "narrator".
However, there is also a literary device, whereby a novel (or similar work that tells a story) is presented as if it were a story told by a person. We call this person the narrator.
The literary narrator's existence is often revealed by the phrasing of the story, because the first person singular is used (and it's not merely quoting a character).
As a very simple example, read the following short story:
I walked to the shop. When I got there, I saw that the shop was on fire. I called our fire department, and the operator picked up the phone. When I told him the shop was on fire, the operator said "I am already aware of the fire".
The narrator is "I". Although it is not clear who this person is, it is clear that they are the one telling the story.
I also bolded one particular "I". It's very important to note that every unbolded "I" in the story refers to the narrator. But the bolded "I" does not refer to the narrator! It's merely a direct quote from the operator, who is not the narrator of the story.
2. The poem has a literary narrator, and it is not the neighbor!
Notice that the first person singular (which refers to the narrator) does not refer to the neighbor:
A neighbor of mine
That's a fancy way of saying
In other words, the neighbor is not the narrator. The neighbor is the neighbor of the narrator.
This is also why we refer to them as "the neighbor". That description is given to us by the narrator, who is talking about a person who lives next to them (thus describing them as "my neighbor").
I want to highlight something you said in your question:
The poet introduces the narrator as his neighbour ...
This is incorrect. The neighbor is not the narrator of the poem. "The poet" (as used by you in the above sentence) is the narrator, specifically because "they introduce the neighbor", which means that they are telling the story and deciding to introduce the neighbor to the reader of the story.
You have inferred that "the poet" introduces the story. It would be more accurate to say that the narrator introduces the story. The narrator is not always the author.
One example of this is A Series of Unfortunate Events. The author (Daniel Handler), has written the story as if it were written by Lemony Snicket.
- Daniel Handler is the author.
- Lemony Snicket is the narrator
In-universe, Lemony Snicket is a character in the story. Lemony has decided to write a book to talk about the things that happened in his past. Therefore, you find phrases such as "I am telling you this story", in which "I" refers to Lemony Snicket.
Lemony Snicket is the author of the (in-universe) book, in the sense that he is the one who (in-universe) has decided to write the book. But that is different from the real (out-of-universe) person who actually wrote the (out-of-universe) book (Daniel Handler).
3. In the poem, the neighbor tells a story.
More specifically, she tells the story of her having done a childlike thing. This makes the neighbor "a" narrator (= a storyteller), but it does not make her "the" narrator (= the narrator of the poem itself).
Think of it this way: if the neighbor was "the" narrator, they they would be using the first person singular "I". When we directly quote the neighbor, this becomes more clear:
"I lived on a farm when I was a young girl. And while I lived on that farm, I did a childlike thing."
In this direct quote, the neighbor is the narrator, as she is using "I" to refer to herself.
However, this is not true for the poem itself:
A neighbor of mine in the village
Likes to tell how one spring
When she was a girl on the farm, she did
A childlike thing.
Notice how the neighbor is referred to as "she" (third person singular). Even though the neighbor is narrating something in which she is the main protagonist (so she would be telling the story of her childhood, saying things like "I lived on a farm"), the poem describes this story from a third person perspective.
The fact that the neighbor's story is portrayed in the third person, and not in the first person, directly proves that the neighbor is not the narrator of the poem, even though the neighbor is a storyteller (narrator) who is a character in the poem.
4. Summary and direct answers to your questions.
This poem is a story about someone (the narrator) who was told a story by their neighbor.
The neighbor is a narrator (storyteller) character that appears in the poem, but they are not the narrator of the poem itself. There are merely the neighbor of the narrator of the poem.
Who is the speaker in the poem?
It's not quite clear who is telling the story. We only infer the narrator's existence by the usage of the first person singular ("mine").
It's important to notice the distinction between "the narrator" and "the author". However, when there is no explanation as to who the narrator is, most people would automatically assume that the author is also the narrator.
So I guess it depends how you look at it. It's fair to assume that the narrator is the poet themselves, but you should remember that this is an assumption and not conclusively proven.
If it is the poet himself, can this style be an example of the dramatic monologue?
Quite the opposite. Dramatic monologue is defined as
- A single person, who is patently not the poet, utters the speech that makes up the whole of the poem, in a specific situation at a critical moment.
This already directly contradicts your suggestion that it is a case of dramatic monologue if the poet is also the narrator.
However, you should also observe the intention of a dramatic monologue:
- The main principle controlling the poet's choice and formulation of what the lyric speaker says is to reveal to the reader, in a way that enhances its interest, the speaker's temperament and character.
Robert Frost's poem never really provides any exposition about the narrator's character, which means that the poem is not trying to be a dramatic monologue.
I can refer back to Lemony Snicket's A Series of Unfortunate Events. Through his narration, Lemony reveals how sick it makes him feel when he thinks about what the Baudelaire children have been through. Lemony is upset with the world for letting such injustice happen, and that is exactly why he has chosen to tell the reader this story.
This is a case of dramatic monologue, because the narrator is patently not the author, and the narrator's narration is used (by the author, as a literary device) to enhance the story by having Lemony provide commentary to the story as it unfolds.
5. Some examples
You've probably seen The Simpsons. In the show, Bart and Lisa often watch a cartoon called "The Itchy & Scratchy Show", about a mouse (Itchy) and a cat (Scratchy) who fight eachother.
Itchy and Scratchy are the main characters of "The Itchy & Scratchy Show", but they are not the main characters of "The Simpsons" itself.
I often visit my grandfather. Every time I visit, he ends up telling me long and boring stories.
I am the narrator. My grandfather is only a character in my story.