Is there a time limit after which you should cite a working paper as an unpublished paper? I am citing a paper with the latest version being 2005. I checked some recent studies that cite this paper and some reference it as a "working paper" while others reference it as an "unpublished paper".

Is there a time limit or rule of thumb after which a working paper should be cited as an unpublished paper?
In some fields, most papers are referred to as working papers even when they are basically preprints. And often they are explicitly in archives for "working papers," or published in a series of "working papers" (often if they were presented at a university for feedback).

After determining that there is not a newer or published version, I usually cite these papers, regardless of age, as a working paper if it is referred to as such. However, depending on the citation system, there may not be an official way to cite something as a "working paper", and so in that case I often default to citing it as an "unpublished paper." (A similar problem, or individual preferences, may be why different authors choose to cite it in different ways.)

I do not think time should change that classification, because "working paper" is a phrase indicating the general status of the work to those in the field. At the extreme, I might still refer to a "working paper" by someone who retired or died, even though it is certainly not still a work-in-progress. For instance, here is the official BibTex file for the first working paper in the NBER archive, a 1973 work by Finis Welch. The citation has type "Working Paper."