Mature mammalian erythrocytes have all the characteristics of a eukaryotic cell except that they don't have a nucleus, they don't have any cell organelles. Does this mean that erythrocytes are classified as prokaryotic?
No. Nobody considers red blood cells to be prokaryotic, perhaps most importantly because they are part of a eukaryotic organism. Red blood cells begin life with the full complement of organelles, including a nucleus and mitochondria, but our RBCs shed their organelles during maturation. In actuality, though, only mammalian RBCs lack nuclei; other animals' RBCs still hold on to their traditional eukaryotic characteristics.
As an analogy, apoptotic or necrotic cells don't have intact organelles, but they are still considered eukaryotic.
No. Prokaryotes have DNA. Mature RBCs don't. And RBCs cannot divide since they lack DNA.
We're looking for long answers that provide some explanation and context. Don't just give a one-line answer; explain why your answer is right, ideally with citations. Answers that don't include explanations may be removed.
No. A Red Blood Cells is formed in the bone marrow as a fully equipped Eukaryote including all organelles and an enclosed nucleus containing linear DNA. These parts are necessary for the development of the cell, but not necessary for the function of the mature cell. They are shed to allow it to squeeze through small capillaries. A Prokaryote contains a nuclear region of circular DNA which is in direct contact with the cytoplasm. Red Blood Cells never develop a nuclear region containing DNA and can therefore not be classified as Prokaryotic. They are still considered Eukaryotes because they contain Eukaryotic 80S ribosomes as opposed to Prokaryotic 70S ribosomes.
Note that the reason mature erythrocytes are eukaryotic is the same reason snakes and dolphins are tetrapods. "Eukaryote" and "Tetrapod" are clades, i.e. groups defined by common descent, which leads to pervasive similarities within the group and differences with organisms outside the group which justifies defining that as a group in the first place, even if we didn't know they were descended from a common ancestor.
That is an important fact about life that is obscured with classifications like "mammals are animals that have fur, bear live young and feed them milk". We could imagine a situation where we could list traits ("has fur"/"has feathers"/"has scales", "swims"/"runs"/"flies", "bears live young"/"lays eggs", etc) and we would find creatures with any combination of those traits. If you knew an animal had fur, that wouldn't tell you what any of its other traits were, because the combinations were endless. At that point, "it feeds milk to its young... but it has no fur and swims!" might lead to big questions of "mammals feed milk to their young, but they also have fur, so should we call this a mammal?". But words like "mammal" would also be a lot less useful, because knowing an animal had fur, gave birth to live young it fed with milk wouldn't actually tell you anything more about the animal in question, since its other traits could be anything.
That is NOT how it turns out living things are. Traits are distributed in a very specific way: a nested hierarchy, groups within groups that don't overlap. This was found by Linneaus, the first person to try and classify all of life. He found that when looking at the similarities and differences between animals, they naturally fell into groups within groups; you had mammals and birds and reptiles and fish, and within mammals you had primates and carnivores and whales, and within carnivores you had canines and felines, etc... with no creature that was, say, half like a dog and half like a duck to the point there was a question of whether it fit in "mammals" or "birds".
This classification isn't arbitrary. Linneaus admittedly did have a thing for the nested hierarchy, and tried to classify rocks the same way. That didn't work out; today's biologists have changed Linneaus' classifications a lot, but the nested hierarchy has proved an enduring fact of how the traits in living things are distributed. On the other hand today's geologists classify rocks completely differently, because it turns out the traits of rocks are not distributed according to a nested hierarchy.
Moreover, biologists have since discovered the cause of this nested hierarchy, which is common descent with modification. Organisms reproduce and have offspring that are slightly different from them; over time species differentiate into different groups that are slightly different from the parent species and those groups become species and differentiate in turn. As a result, how similar two organisms are to each other is directly related to how recently they share a common ancestor. Groups emerge within groups because they descend from one another, and don't overlap with far-away groups because there is no mechanism for that to happen (main exception being bacteria, who can swap genes with other bacteria from very different groups, and as a result they don't fall in as clear a nested hierarchy as eurkaryotes do).
This means that there is a difference between modern biology and that of Linneaus's time; Linneaus listed visible traits and classified organisms based on them. It so happened that they fell into a useful nested hierarchy, so "has fur and feeds milk to its young" told you a lot more about the animal, like "has lungs, has a backbone, probably has four legs", etc, but the list of traits was still the main important thing. Nowadays we know of a lot more traits; we know an organism's genes, the structure of its proteins, details of their physiology that biologists of Linneaus's time could only guess at. While in Linneaus' time you might be able to say "dolphins have no fur and swim like fish, but they have lungs like mammals and bear live young; where to classify them? I guess they're kinda more like mammals?", today we know that dolphins are overwhelmingly like mammals down to the molecular level, in literally every way other than "they have no fur and swim". And we also know the reason why: dolphins share a common ancestor with all other mammals dating back to the Triassic or so, while their common ancestor with fish dates back to the Ordovician, hundreds of millions of years earlier.
This means that today's organisms are by and large classified on the basis of common descent. The "Eukaryotes" group certainly is (it's a clade). What makes an Eukaryote an Eukaryote today isn't technically any specific trait, it's whether it descends from the common ancestor of Eukaryotes or not. Red blood cells absolutely meet that criterion. But this isn't just an arbitrary classification choice; it so happens, because of this common descent, that red blood cells are a lot more like other eukaryotes than they are like any prokaryote. "Has/Lacks a nucleus" is one single trait; an important one, important enough to make it the name of the group, but it's not the only one. And in their proteins, ribosomes, metabolic processes,... really everything BUT "does it have a nucleus", red blood cells have more in common with Eukaryotes than with Prokaryotes, so it makes absolute sense that they would be classified that way. Same reason snakes and dolphins are "Tetrapods" even though they don't have four legs; four legs is an important trait, it named the group, but it's still only one trait. "It's almost totally like a lizard, but... it has no legs!" is not actually a riddle or a stumper; "almost totally like a lizard" trumps "number of legs" any day.
How this intellectual deception works
There have been several admirable comprehensive answers to this question, but, having seen it come up again recently, I thought it would be useful to point out why something like this that is clearly wrong can sound plausible. In general there is some sleight of hand involved.
In this case the sleight of hand is the (deliberate?) confusion of ‘cell’ with ‘organism’.
Prokaryotic and eukaryotic are terms that apply to organisms. Prokaryotes are unicellular, so a prokaryotic organism is often loosely referred to as a prokaryotic cell. Some eukaryotes are multicellular, and multicellular eukaryotes such as those containing erythrocytes have a variety of different specialized cells, but these cells are not organisms. The interchangeable terminology for prokaryotes does not apply here, and, as one always knew, it is nonsense to talk about particular cells being prokaryotic or eukaryotic.
The fact that nuclei are a characteristic of eukaryotic organisms does not require every cell of a eukaryotic multicellular organism to have a nucleus (and anyone who tries to define it that way is wrong.)
Don’t look where the magician wants you to look.