So it’s worth prefacing my comment by noting that this kind of thing has popular traction because the model of the history of science that most people have in their heads is wrong. The “textbook history of science” model (as we historians of science refer to it) is a march of “innovations” attached to “heroes” and usually is meant to impart to scientists (and perhaps citizens) some kind of moral message about how science works.

So in my astronomy course as an undergraduate, the history was more or less represented in the following heroic sequence: Copernicus (comes up with an unintuitive but striking theory), Brahe (makes lots of systematic observations to build up a data store), Kepler (takes the data and comes up with some radically useful equations to describe nature), Galileo (shows the value of new instruments, and defends the theory against extra-scientific forces), Newton (synthesizes it all into a coherent, grand theory), Einstein (overturns the slightly-wrong theory with a new, highly unintuitive theory, inspired by the data from a crucial experiment, changes the entire world, etc.).

This is a highly impoverished version of history, even if it serves the need of the scientist. It is “modeling” an ideal of science as a mixture of theory, observation, experiment, instrument, and autonomy, with Einstein in particular serving as the reason for the whole thing (changing the entire world). If you study science, this is not just a nice story to tell about your noble profession, but it is a way to explain to non-scientists the intellectual honesty of scientists and the importance of funding science and letting scientists do their thing with autonomy.

But a historian would tell you that these pocket descriptions are all not quite right, and that of course there were literally thousands of other people who were crucial to the development of any theories one might care about today, and certainly many more people, contexts, institutions, etc., that were crucial to the developments in question. And there were lots of people who were absolutely necessary for these “big names” to exist — e.g., everyone knows Galileo, but nobody other than historians know the names of the people he was engaging with directly in his work, the Jesuit astronomers like Clavius and Scheiner. Galileo was not working in a bubble. (And, indeed, in Galileo’s case, the lack of a bubble is significant — he is a famous case of context biting back.)

The stakes for this question is that the “Einstein was a plagiarist!” narrative (which goes back to the anti-Einsteinians of the 1920s and 1930s) appeals to people with bad models of history. Nobody but some specific branches of historians and mathematicians know about the work of Poincaré and Lorentz, and so when people find out that Einstein’s work is not totally “out of the blue,” people get very excited and interested. It seems to be overturning the (impoverished) historical narrative, and especially people who do not study fin-de-siècle physics, the differences between what Einstein was doing and what others were doing is not apparent, and so it looks like basically the same thing.

Historians of physics have long, long known that Einstein was not working in a bubble. Intellectual or otherwise. Considerable historical work has been done on reconstructing what other thinkers influenced Einstein, and who he thought he was engaging with when he published his papers (it was not “the general public”). Some of the most important people that Einstein saw his work as intersecting with was Ernst Mach (the physicist-philosopher whose philosophy of methodological positivism influenced Einstein’s kinematical worldview), Henri Poincaré (whose work on questions of simultaneity and electrodynamics touched on core questions for Einstein), and Hendrik Lorentz (whose attempts to create a coherent aether theory that accorded with the inability to the detect the aether inadvertently gave Einstein some of the mathematical tools necessary to dispose of the aether), to name a few. To say nothing of the work by James Clerk Maxwell, which of course provided the electromechanical “backbone” for all of these kinds of forays. It is of note, however, that while he was working on his early theory, Einstein’s connectivity with these theorists was limited — he had read some of their works, but was not “in the weeds” with everything they had output.

Poincaré, Lorentz, and others had various theories that we could lump under the idea of relativity (and the basic concept of relativity, by itself, is much older than Einstein — you can think of Einstein’s relativity as being what happens if you apply the principles of Galilean relativity with the notion of a fixed speed of light and Maxwell’s equations; special relativity just “falls out” of that attempt to make all of these things be true at the same time). But their approaches were very different, and their goals were very different. Neither Poincaré nor Lorentz ever believed that relativity was totalizing — “local time” could be relative, but they still believed there was some form of “universal time,” something related to the aether itself (which they still believed in, to various degrees of ontological realism). To these guys, Einstein was answering a different question, and doing it backwards: for Einstein, you had to get your philosophy and kinematics (philosophy of measurement, i.e. “time is what clocks measure”) sorted out before you tried to figure out how electrodynamics worked, for Poincaré et al., it was exactly the other way around. As the historian of physics Helge Kragh puts it (in Quantum Generations):

Poincaré's theory was an important improvement, a relativity theory indeed, but not the theory of relativity. Strangely, the French mathematician did not follow up on his important insights, nor did he show any interest in Einstein's simultaneously developed theory of relativity.

Which is to say, in basic terms: yes, there was conceptual and even mathematical overlap between Einstein, Poincaré, Lorentz, et al. There were also significant conceptual and even mathematical differences. To say that “Einstein plagiarized Poincaré” is to not understand either Einstein or Poincaré (Poincaré would never have thought this, because his work and Einstein’s work was, in his eyes, totally different).

Einstein’s 1905 papers are very strange to modern eyes, in part because their citation style is indifferent. I will just note that citation practices vary over time (and varied dramatically over the case of the 20th century), but it is also of note that Einstein was writing these papers not as a university academic, but as a patent examiner. Indeed, the historian of physics Peter Galison points out that:

It has long struck scholars that the style of "On the Electrodynamics of Moving Bodies" does not even look like that of an ordinary physics paper. There are essentially no footnotes, very few equations, no mention of new experimental results, and a lot of banter about simple physical processes that seem far removed from the frontiers of science. Pick up a typical paper from the Annalen der Physik and a very different form appears in nearly every article: they typically begin with an experimental problem, a calculational correction; they are filled with references to other papers. But read Einstein's paper through the eyes of the patent world and suddenly it looks not so strange-at least in style. As one recent author has commented, patents are precisely characterized by their refusal to lodge themselves among other patents through footnotes-that would compromise the entrepreneurial advantage the author seeks. The simplistic banter is not unusual; patents are in fact written for a "person skilled in the art" (as patent language had it), not specialist readers. And the story told by the author aims to describe a procedure rather than point the way in a long train of work.

(Galison, “Einstein’s Clocks: The Place of Time,” Critical Inquiry 26, no. 2 (200), 355-389)

Anyway. No historian of physics has any truck with the Einstein plagiarist story. It’s at best a misunderstanding of the history, at worst a deliberate misreading of it. Why “deliberate”? Fame-seeking, perhaps — there are stakes to “overturning Einstein.” There have been nationalist reasons — pro-French, for example — for elevating Poincaré, whose work, to be sure, has been dramatically underappreciated by Anglophone audiences. And, historically (and not everyone who advocates such things today is in this camp, I will emphasize), there have been deeply ugly motivations for attacking Einstein and his work (explicitly anti-Semitic reasons). There is a small anti-Einstein industry (to be sure, much smaller than the massive pro-Einstein industry, which has its own “heroic” problems) that is variously motivated, but the basic argument becomes summed up as “Einstein’s work was wrong, and he plagiarized it anyway!” It is not historically accurate (and it also presumes that pretty much all physicists today are total idiots... which is a pretty big assumption), it is regarded as crank history.