There are several important aspects to your question.
Firstly it would be good to know what you would understand by modern science. Psychoanalysis is clearly outside the range of physics, mathematics or computer science. If you look deeper into it - so is an awful lot, if not all, of positivistic psychologies (especially when applied to psychological therapies) that are currently considered a valid field of study at university level.
What psychoanalysis is first and foremost, in my opinion, it is a clinical discipline. Psychoanalysis is not something that you can learn from books or Wikipedia; there is not even such a thing as a universally recognised textbook of psychoanalysis. But there are such textbooks in biochemistry, physics and mathematics. Furthermore - writings from various psychoanalytic authors tend to look at similar presentations to a significant extent differently.
Hence the answer to your question is rather complicated. The concept of projection is alive and well within the field. It is part of the language of psychoanalysis; language that is being used to describe human experience and as such it has a clear face validity. It is a term often also used within psychiatry and in practice of non-psychoanalytic psychologies. On the other hand presence or absence of the phenomenon of projection is equally clearly a subjective decision. Here your expectations of what modern science is are important.
Lastly I would like to mention that insights from biology and neuroscience do matter. They are very important and do add to our understanding of human brain and mind. But as for practical aspects of psychological therapies of any kind that I know of they are relevant mostly as a tool for triangulation between biological descriptions and psychological. When I first started in this rather complicated field I had an expectation of the great importance of neuroscience to the practice of psychological therapies. The insights are important, of course, but not really for the practice except as far as they change what the therapist thinks of the human condition. At least for now and, sadly, I cannot see it being more than that for the next decades. And please do not get me wrong I firmly believe that psychoanalysis itself would benefit from more research, perhaps especially on the information that is provided by the patients.
One could say then, of course, does this archaic practice of psychoanalysis or therapies derived from it matter? Well, it does matter a lot, the moment you or your relative develop serious enough psychological problems to think that life is not worth living or you feel trapped in problems that just seem to hit you again and again. Depending on your predilection you may seek phrmacological help and/or psychological.
In terms of references I could offer several kinds: there are studies about what kind of psychological therapies professionals choose for themselves; there are outcome studies for various psychological therapies. Then there are articles directly on the subject of psychoanalysis as a scientific theory. None could however provide a direct and valid answer to your question.
It seems that your question is sparked by Arnon Weinberg's answer to the question: Is there a term that describes the behaviour of a person which transfers personal problems to other people and tries to solve them there? That question describes a clinically defined problem and Weinberg's answer is clinically valid. I think it may be relevant to direct also to the famous stackoverflow answer that starts with: You can't parse [X]HTML with regex.
