Short answer: No, mushrooms are NOT plants. Period. They are the spore-bearing structures of fungi, and fungi are a completely separate kingdom of life from plants.
Long pontification follows, about taxonomy…
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Once upon a time, I think this may have still been true at the early parts of my own lifetime, all forms of life were considered to be either plants or animals.
By some time, at least in the late 1960s or early 1970s, we knew that taxonomy was more complex than that. I’m pretty sure, by then, that we knew that fungi were not plants, but a kingdom of their own.
When I was in college, in the 1980s, I learned the Five Kingdom system. An instructor drew a diagram on the blackboard that looks like a Micky Mouse hand with three fingers. I hoped that if I Googled, I might find an image that looks like my Botany professor’s Mickey Mouse hand drawing. Not quite, but if you look at this image, maybe you can get the idea, and make sense of what I will hereafter write.
The three fingers were the three multicellular kingdoms of life, animals, plants, and fungi.
The palm of the hand was the kingdom Protista, which supposedly included all unicellular eukaryotic (meaning that they had a complete cell nucleus and organelles) forms of life. Except yeast. Yeast, I was taught, were fungi, even though they were unicellular. Thinking back, I realize that many other life forms, classified into Protista, were really similar enough to plants or animals, just as yeast was to fungi, that they probably belong there as well; and I am pretty sure than under more modern systems of taxonomy, they are.
The wrist was Monera, simpler unicellular life forms, prokaryotic (meaning that they did not have nuclei or organelles). Two forms of life, I was told, were included in Monera, bacteria, and blue-green algae.
Anyway, under the original two-kingdom model, fungi were indeed considered plants. One of the things driving taxonomists to decide that that model was too simple was surely the realization that fungi are at least as different from plants, as animals are from plants. In fact, I believe more modern genetic analysis has shown fungi to be more closely related to animals, than to plants.
Much more recently, I have learned that under a new system of taxonomy, kingdoms are not even the highest taxon, any more. Now, above kingdoms, are three domains—Archaea, Bacteria, and Eukarya.
Eukarya includes everything that was under plants, animals, fungi, and protists. Plants, animals, and fungi are still recognized as kingdoms within Eukarya, but many life forms no longer fall under any defined kingdoms.
The name “blue-green algae” is now deprecated. They are no longer considered to be any kind of algae, but are now considered a form of bacteria, and called cyanobacteria.
The big discovery was that certain organisms previously thought to be a weird sort of bacteria, are really something else entirely. Different, enough, not to merely call for a Kingdom of their own, but for a new taxon above that; hence the domain Archaea. There have been some speculations that whatever event first created life on this planet, that there might have been two separate events, with Bacteria having been created by one, and Archaea by the other. I think that theory is generally not held to be true, but the point is that Archaea are that much different from Bacteria to give rise and credibility to such a theory.
And I believe that Eukarya are now most widely thought to have evolved from Archaea, not from Bacteria.