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I am a PhD student and I wrote a paper in the social sciences. The contribution of my paper is that it's the first non-North American analysis on the subject.

There is a working paper I found online from a doctoral student in another university, whose research is semi-similar to mine, and who looks outside of North America.

I am submitting my paper to an academic Journal. I have not cited the other students paper. It's been two years since they uploaded the working paper, and there is no sign of it published anywhere, and no update on the students (now a post doc) own research website. I tried to contact the author to ask if they intend to publish, I got no response.

I did not reference their paper in my research, because I don't think it's right that I should forfeit my contribution for an incomplete paper that doesn't seem to be heading to a journal and undergo peer review.

I am seeking opinions/guidance as to whether I am right. My supervisor has provided no guidance, and I can't find information on the internet.

I think in your literature review where you point to the gap in studies that look outside of North America, you could add something like, "Little research has applied this theory/concept/idea outside of the North American context (for an exception, see Smith, 2016).

And then cite it in your references as
Author, A. A. (2016). Title of paper or manuscript. Unpublished manuscript. Date retrieved. URL.

You need to cite the paper only if you use it or depend on it in your research in some way. However, if you have a section on "Similar Research" then you should probably mention it there. You don't forfeit anything if your work was independent. For purposes of your degree, your supervisor will decide if you have made a proper contribution. For purposes of publishing, the editors and reviewers will decide on priority. Just because you don't cite it doesn't mean that it isn't known.

Independent research on a topic goes on constantly in the real world.

The worst case for you is that if you don't cite it and someone thinks you have plagiarized. But the real world situation is as it is. You can't change that.

I'll add another aspect to the main issue that's covered in Buffy's answer. It seems that the working paper is not at all identical to your work. As you say, it is "semi-similar" and focuses on a different region. First, this means that your paper remains original. Second and more importantly, your work and that of your colleague seem complementary. You said you already contacted your colleague without response, but in general, this is a great opportunity for collaboration, for example on a comparative paper. Maybe you'll get a response if you suggest a collaborative project that builds on both of your contributions.

New answer

Comments below explain the OP posted their work online first and that their underlying question is about priority, in particular, is priority awarded to the first author(s) to make their work public or the first author(s) to formally publish their peer-reviewed work? My original answer to the original question appears below, and my answer to the aforementioned question is as follows:

Priority is rather contentious. Ultimately, no one can deny that a work publicly available before another takes priority, regardless of whether one work is formally published after peer-review and the other was merely part of the public record. Moreover, no one can deny that authors of the second work had an advantage. Nonetheless, some will argue that being first doesn't matter; being formally published does. But, they still cannot deny the previous statements. Thus, they can only make claims such as "this is the first published, peer-reviewed work that ...," which implicitly acknowledges that the work isn't first. I recommend avoiding such claims altogether and mentioning related work, regardless of whether it is formally published.

Original answer

You open by explaining

...The contribution of my paper is that it's the first non-North American analysis on the subject.

Yet,

There is a working paper...from a doctoral student in another university...who looks outside of North America.

So, your claimed contribution is seemingly a lie: It's [not] the first non-North American analysis on the subject.

(I don't understand how a paper can contribute simply by conducting analysis in a new geographical region; to contribute, you surely need to bring new science. I'll proceed regardless.)

I did not reference their paper in my research, because I don't think it's right that I should forfeit my contribution for an incomplete paper that doesn't seem to be heading to a journal and undergo peer review.

I consider that to be morally and ethically wrong. You should rightly forfeit your claim of first non-North American analysis, because by your own admission, you aren't first.