In general, a lot of terms in literature are not very strictly defined: the definitions may vary among different schools of study and different people may use the same term in different ways. I say this not only from my viewpoint as a mathematician (coming from a field where every term is very strictly and unambiguously defined) - it's quite clear in discussions about the definition of literature itself (about which this site has a whole tag) such as this answer, as well as attempts to define specific genres, that literature is a topic where most terms do not have a single fixed meaning that everyone agrees on.
I mention this in order to clarify that this answer is not completely debunking your proposition that a stream of consciousness should not have punctuation or line breaks. The example cited by Wikipedia from James Joyce's Ulysses certainly has no punctuation, and Wikipedia says, "Stream-of-consciousness writing is usually regarded as a special form of interior monologue and is characterized by associative leaps in thought and lack of some or all punctuation."
a quarter after what an unearthly hour I suppose theyre just getting up in China now combing out their pigtails for the day well soon have the nuns ringing the angelus theyve nobody coming in to spoil their sleep except an odd priest or two for his night office the alarmlock next door at cockshout clattering the brains out of itself let me see if I can doze off 1 2 3 4 5 what kind of flowers are those they invented like the stars the wallpaper in Lombard street was much nicer the apron he gave me was like that something only I only wore it twice better lower this lamp and try again so that I can get up early
This is what we might call an extreme form of the stream-of-consciousness style of writing. Note the "some or all punctuation" - it's not necessary to dispense with full stops and dashes altogether. Let's check another definition, from LiteraryDevices.Net. Here it says, "The stream of consciousness style of writing is marked by the sudden rise of thoughts and lack of punctuation." But look at the examples. Firstly, again from Ulysses:
He is young Leopold, as in a retrospective arrangement, a mirror within a mirror (hey, presto!), he beholdeth himself. That young figure of then is seen, precious manly, walking on a nipping morning from the old house in Clambrassil to the high school, his book satchel on him bandolier wise, and in it a goodly hunk of wheaten loaf, a mother’s thought.
Secondly, from Virginia Woolf's Mrs Dalloway:
What a lark! What a plunge! For so it always seemed to me when, with a little squeak of the hinges, which I can hear now, I burst open the French windows and plunged at Bourton into the open air. How fresh, how calm, stiller than this of course, the air was in the early morning; like the flap of a wave; the kiss of a wave; chill and sharp and yet (for a girl of eighteen as I then was) solemn, feeling as I did, standing there at the open window, that something awful was about to happen …
Thirdly, from David Lodge's The British Museum is Falling Down:
It partook, he thought, shifting his weight in the saddle, of metempsychosis, the way his humble life fell into moulds prepared by literature. Or was it, he wondered, picking his nose, the result of closely studying the sentence structure of the English novelists? One had resigned oneself to having no private language any more, but one had clung wistfully to the illusion of a personal property of events. A find and fruitless illusion, it seemed, for here, inevitably came the limousine, with its Very Important Personage, or Personages, dimly visible in the interior. The policeman saluted, and the crowd pressed forward, murmuring ‘Philip’, ‘Tony’, ‘Margaret’, ‘Prince Andrew’.
All of these use at least some punctuation. As for line breaks, well, they're one of the main things distinguishing poetry from prose. Without line breaks, a poem wouldn't be a poem any more. So if we want the phrase "stream-of-consciousness poem" not to be an oxymoron, then line breaks must be allowable too. And certainly some critics have referred to stream-of-consciousness poems: for example, T. S. Eliot's "The Love Song of J Alfred Prufrock" has often been referred to as employing a stream of consciousness technique, skipping from one barely-related thought to another, despite the punctuation and line breaks:
I grow old ... I grow old ...
I shall wear the bottoms of my trousers rolled.
Shall I part my hair behind? Do I dare to eat a peach?
I shall wear white flannel trousers, and walk upon the beach.
I have heard the mermaids singing, each to each.
I do not think that they will sing to me.
TL;DR: the definition of what constitutes a "stream of consciousness" isn't set in stone, but at least some definitions certainly allow the existence of punctuation and line breaks.