The mathematical reason for how parallel universes exist is obviously complicated, so people referring to "dimensions" as an alternative of "parallel universes", "realms", "other worlds", and the like, seems to be doing so out of confusion and improper research. Unfortunately, this usage proved contagious and has become a trope, so now seemingly every sci-fi and fantasy author believes that "dimensions" means "other worlds". Let's discuss some factors that might have led to this confusion.
Types of multiverses
There are many theories on how parallel universes can exist, and these two are some of the most popular in my opinion:
- Brane cosmology from string theory:
The brane multiverse version postulates that our entire universe exists on a membrane (brane) which floats in a higher dimension or "bulk". In this bulk, there are other membranes with their own universes. These universes can interact with one another, and when they collide, the violence and energy produced is more than enough to give rise to a big bang. The branes float or drift near each other in the bulk, and every few trillion years, attracted by gravity or some other force we do not understand, collide and bang into each other. This repeated contact gives rise to multiple or "cyclic" big bangs. This particular hypothesis falls under the string theory umbrella as it requires extra spatial dimensions.
- Many Worlds Interpretation of quantum mechanics (arguably the most popular, but has nothing to do with higher dimensions on its own):
The quantum multiverse creates a new universe when a diversion in events occurs, as in the real-worlds variant of the many-worlds interpretation of quantum mechanics.
Types of multiverses: https://en.wikipedia.org/wiki/Multiverse#Types
Brane cosmology is a relatively new theory (first documented application in 1983) and might not be the main cause of the confusion between "dimensions" and "worlds", but it is directly related to a type of multiverse that has inspired more recent literature on parallel universes. Going further back in time, the cosmology of Einstein's General Theory of Relativity (1916) proposed a higher fourth dimension, which likely inspired earlier authors. However, the reasoning may be the same, in that writers may conjecture, however romantic, that entities may travel to parallel universes via a "path" through higher dimensions. E.g., the fantastical act of a flatlander in a 2D world jumping to a different 2D world via the third dimension like a drawn stick figure jumping off his home paper on your desk and over to the next paper (however that would look or work). At least this conjecture offers some sense of mechanics and attempts to distinguish "dimensions" from "other worlds", and that "dimensions" was a property of those worlds and not those worlds itself. I.e., a 3D world is still just a world and not "a dimension". But, as we know now, the term "dimensions" has been bastardized further to become these other worlds itself, and not "mediums" or "paths" to them. The reason for this may be because when people heard that a flatlander traveled through the "third dimension", people thought of the third dimension as a world. People also heard how the flatlanders lived in the "second dimension", which is also thought of as a world. But the problem with this usage is the conflation of property of that world, its dimensionality, and the world itself. The third dimension isn't a "world", but a property of worlds. E.g., the two separate pieces of paper that a flatlander can live in are two different worlds sharing the property of two-dimensionality. Or in the Many World Interpretation, the universe where the cat is dead and the universe where the cat is alive in the Schrodinger's cat thought experiment are two different universes having the same dimensions. Sci-fi literature that do make attempts to distinguish "dimensions" from "worlds" still often contain potentially confusing passages that conflated "dimension" and "world" (for an example, see an excerpt from an article in a later section of this comment). So, it's not difficult to see how people naturally started to think "dimensions" equals "worlds".
Paths through higher dimensions in literature: https://en.wikipedia.org/wiki/Fourth_dimension_in_literature
Pseudoscience
It was difficult finding a resource directly answering the question, and perhaps this is why people continue to confuse the terminology, which is also the reason I decided to make this post. There is an article from the University of Texas physics department saying that popular pseudoscience may have a hand in the confusion of terminology (it's worth reading the whole article for a short historical understanding of how physicists or mathematicians actually use "dimensions"):
Beginning in the last decade of the 19th Century, newspapers and
magazines were full of completely confused accounts of "the fourth
dimension," not to mention "vibrations" and "energy," two hot topics
of late 19th Century physics. When Einstein came along, the "fourth
dimension" became an even hotter topic, and acquired some curves too.
Higher dimensions in pseudoscience are often even further confused
with then-unrelated scenarios, such as "coexistent worlds," "parallel
worlds," the "worlds" reached in dreams and drug-visions, not to
mention Heaven, Hell, and even other planets. Mindless journalistic
publicity for string theories has over the past two decades touched
off yet another wave of science-fictional and fantastical delirium
involving "multiple universes," a delirium owing a lot to late 19th
Century Theosophy, but essentially nothing whatsoever to advances in
physics.
Higher dimension pseudoscience: https://web2.ph.utexas.edu/~coker2/index.files/4d.shtml
Spiritual Dimensions
There is also a usage of "dimension" in the term "spiritual dimension", which has nothing to do with "parallel universes", "worlds", "realms", or the like. Spiritual dimension are the dimensions of the mind. E.g., dimensions of 3D space is length, width, and height, but the dimensions of spirituality might be "consciousness", "purpose", "meaning of life", or the like. I was compelled to add this here because Merriam Webster's fifth definition of "dimension" is "a level of existence or consciousness", referring to the spiritual dimensions. But I can see how someone may be confused and take "level of existence" to mean other worlds.
Spiritual dimensions: https://pubmed.ncbi.nlm.nih.gov/8550306/
Merriam Webster definition: https://www.merriam-webster.com/dictionary/dimension
When Was The First Usage?
It is likely owed to no single author in popularizing the wrong usage. As for first usage, a lot of the older, shorter literature is lost, so I can only make an approximation. Algernon Blackwood's "The Pikestaffe Case" (1924) was the earliest sci-fi story I found where "dimensions" referred to worlds.
So this mathematics professor ("Mr. Thorley is a high mathematician. He makes measurements and calculations") who reads "Gauss! Minkowski! Lobatchewski! Einstein!" takes up lodging at an old woman's guest-house and starts performing some experiments with an antique mirror in the house (going so far as to order "mathematical instruments"). Then, one day, he disappears and so does his best math student, Gerald Pikestaffe. The old lady sees them hovering in some new dimension inside the mirror, faints and later, predictably, the mirror breaks and all links are lost.
I also found an article in a "Wonder Stories" magazine from 1931 that seemed to have the usage that dimensions are mediums or paths through worlds or time, or that extra dimensions are some property of reality overlapping the current universe that cannot be observed by humans, but which a latter usage in the same story could be interpreted by the reader as "dimensions" being "worlds".
There was also another race on Earth which was not human but equivalent in every way to man, though utterly different from mankind in its aims, makeup and development. I could not quite make them out as they were absolutely beyond my experience. But I was told that there were many more such races in the universe, though in other dimensions.
The passage technically distinguishes between "dimensions" and "universes", but instead of "universes", it still implies there is some magical realm overlapping the current universe as a separate world. The way it says "in other dimensions" effectively conflates this magical realm with the term "dimensions". Despite this, neither of those works were probably as impactful as later works by the likes of H.P. Lovecraft's which used "dimensions" to refer to "worlds" or "realms", such as in the following passage from Lovecraft's "Dreams in the Witch House" (1933):
...for who could foretell the conditions pervading an adjacent but normally inaccessible dimension.
Here, Lovecraft directly referred to the property of dimensionality as a world that can be accessed, thus conflating the world itself with the dimensionality of that world.
Whether or not these authors themselves knew that "dimensions" did not mean worlds, the confusing usage of "dimensions" in these earlier works likely influenced later authors' treatment of the word "dimensions" as entirely other worlds. E.g., in Cassandra Clare's "The Mortal Instruments", dimensions basically mean "other worlds" and demons come from the "hell dimension", and thus here, the notion of dimensionality as a property is entirely lost.
Further reading:
sf-encyclopedia.com/entry/dimensions
sfdictionary.com/view/1241/dimension
Conclusion
In conclusion, "dimensions" still means the axes in a coordinate system. But unfortunately, due to earlier confusing usages predominantly in sci-fi and fantasy writing, "dimensions" seems to have become an acceptable stand-in for other worlds, universes, or realms in the sci-fi and fantasy sub-genre (I pray it doesn't spread elsewhere). It is a pet peeve of mine whenever I come across "dimensions" in a sentence when what they're really trying to say is "other worlds", and I hope writers begin to move away from such usage.
