Do I mention that a paper in mathematics is a "Short Communication" or "Note" or "Research Article" when citing it?

No, you should NOT include this information. These designations are journal-specific and would provide little information to someone not familiar with each of the journal's classification schemes. For example, a short communication in one journal might represent a more impressive paper (because for some journals a very short paper might be, on average, more impressive than that journal's average quality paper [1]) than in another journal where the bulk of the papers might be short papers, often being "announcement of results" type papers. Also, many journals have no such classification, and thus the reader of your bibliography may wonder whether the paper was not classified in this way by the journal or whether you simply forgot to include the journal's classification.

[1] "Shorter Notes was established to publish very short papers of unusually elegant and polished character ..."

No. The citation guides I know of do not distinguish between types of published contents in journals. They simply use the overarching term of an "Journal Article" (e.g. APA) and do not require one to note whether it's an Editorial, a Review, a Research Article or Short Communication (or whatever of the ca 45 labels there are - possibly with the exception of a Retraction).

Perhaps the underlying reason is that there is not so much of an added value to the reader how the journal happens to categorize an article.

In my field, there is no distinction. Short and long papers are cited the same way.

However this may vary by field. To find out if it is important to your specific field, read papers in your field. Do they make any distinction when citing? For example, after you've written your manuscript, take a look at all of the references you cited. How do they cite their references? Also figure out what journal or conference you plan to submit to. Read some papers on that journal or last year's conference. You should use a similar style.