Yes, the D&D world identifies mushrooms as plants
Historically, the word "plant" referred to both green plants and mushrooms. The microscopic fungi responsible for certain diseases wouldn't have occurred to people as something that could exist until the germ theory of disease gained prominence (late 19th century). D&D means "plant" in this older sense.
This can be hard to stomach. I enjoy D&D in part for the mechanics; I like picking apart the details of how things work and figuring out whether a given mechanic applies in a given situation.
Within the last several decades, the very intelligent people who make careers out of studying plants and fungi reached an overwhelming consensus that fungi are actually more closely related to animals than to plants. They obsessed over the finer details of how things work, and they figured out that plants and fungi are governed by wholly different sets of mechanics, to the point where it's just wrong to call a fungus a plant or vice versa.
Back up, though. Spells like "Speak with Plants" don't say "members of the plant kingdom". They simply refer to "plants". The Player's Handbook is a non-scientific document. It never says what definition of "plant" to use. Several things about the 5e rules point to an older, less-informed understanding of plants.
Consult the Monster Manual's definitions of creature types.
Plants in this context are vegetable creatures, not ordinary flora.
Most of them are ambulatory, and some are carnivorous. The
quintessential plants are the shambling mound and the treant. Fungal
creatures such as the gas spore and the myconid also fall into this
category (Monster Manual p. 7).
I'll point out that, by a scientific definition, none of the monsters listed are plants or fungi. As we understand them in the real world, plants and fungi lack muscles and central nervous systems. A scientist would see a shambling mound or myconid and wonder, "How does it move? How does it make decisions?" Many research projects later, the scientific community would revise its definitions (with the details depending on what research revealed about how these creatures functioned).
There are two takeaways here. First, you're just looking for trouble if you try to get too scientific about what words mean in D&D. Second, this is an example where the rules explicitly don't make a distinction between plants and fungus.
This doesn't fully answer your question, though. Speak with Plants targets ordinary flora, whereas the quoted passage explicitly says it's not talking about ordinary flora.
We don't have the right word
Pretend I'm right-- "Plants" in the spell Speak with Plants refers to the non-locomotive, macroscopic organisms attached to the surfaces of wilderness areas (as well as some other places). What's the correct catch-all word for that? If you limit yourself to scientifically-precise words, I don't think we have one. I think this is a case where, instead of taking half a page to write out what does or doesn't count as a plant, the rules-writers figured it would be less clunky and more fun to use the vague term and let the DM exercise discretion.
How In-Game Characters Would See It
D&D takes place in a bygone technological era, back before people distinguished between plants and mushrooms.
You arm your troops with swords, pole arms, and bows. Full plate armor is still the best defense available. There are no steam engines; coal is used to heat stuff, not to make machine parts move. Assume a parallel level of understanding of mycology. From the Wikipedia page:
Historically, mycology was a branch of botany because, although fungi
are evolutionarily more closely related to animals than to plants,
this was not recognized until a few decades ago.
The Greek philosopher Theophrastos of Eresos (371-288 B.C.) was perhaps the first to try to systematically classify plants; mushrooms were considered to be plants missing certain organs.
The Middle Ages saw little advancement in the body of knowledge about fungi. Rather, the invention of the printing press allowed some authors to disseminate superstitions and misconceptions about the fungi that had been perpetuated by the classical authors.
The start of the modern age of mycology begins with Pier Antonio
Micheli's 1737 publication of Nova plantarum genera.
Sure, you could declare that yours is a world where people actually do understand the difference between plants and mushrooms. By default, though, I think it's understood that characters in D&D have a pre-industrial-revolution understanding of science.
How Gary Gygax et al. saw it
In a similar vein to the previous point, it's unclear to me whether the "Mushrooms aren't plants!" idea would have reached the writers of the first-generation AD&D source books. Sure, 5e isn't 1e, but 5e does draw its inspiration and terminology from previous editions. Calling a fungus a plant feels like a linguistic throwback nowadays, but it's not like the 5e writers invented the throwback from nothing; they borrowed it from an era when it wasn't so much of a throwback.
What's the point of a druid?
Druids love nature. Spells like Plant Growth and Speak with Plants give druids a tangible benefit when they're in wilderness or agricultural areas.
The ecological details of The Underdark are different: Instead of green plants, The Underdark has mushrooms. Despite this, The Underdark still has wilderness and agricultural areas. Does a druid's magic recognize a field of mushrooms as being relevantly similar to a field of barley? I believe the answer is "yes".
The Player's Handbook explicitly says that a Circle of the Land druid can be initiated in The Underdark (PHB p. 68). The Underdark has druids; druids should function properly in The Underdark.