At the bottom of this webpage are publication requirements for PhD students at Skolkovo Institute of Technology. Is it common for PhD programs to have specific publication requirements like this?
I think it's not common to have specific requirements like that on the website, but it is pretty common that something like this exists at least as a guideline in the heads of supervisors and examiners. In every school I have studied / taught at so far in CS, rough numbers of what a minimum PhD thesis looks like were at least floating through the room (and they were not so far off from what is listed explicitly on this website - at least two somewhat beefy papers, usually plus some assorted materials). This isn't yet a good thesis, but it's one that most people would be ok signing off on.
To be honest, it's probably a good idea to write it on the website as this program does - a little bit of transparency often goes a long way.
I think the answer is no, not all do. But if you rephrase it slightly you get a different result. If we define PhD strictly as a research doctorate (though some schools might be weak on that), then a dissertation of some sort is required. In all fields, and for all advisors, I think that the attitude, if not the specific rule, is that the dissertation be of publishable quality. The exceptions, where the committee doesn't have such an attitude, seems to me to be a vanishingly small set.
That is a judgement made by the advisor and committee in many fields. In some fields, and apparently at some universities, prior publication is required. But that is just a way of having an independent judgement about publishable quality; by the journal/conference reviewers. The decision is, then, somewhat distributed.
Some dissertations never get published other than informally (university libraries, University Microfilms, etc). Some result, after the fact, in one or more publications. In my own case (though long long ago) there was no formal requirement other than a submission to what was then University Microfilms. But a shorter version of the dissertation was published a few years later.
My US institution does not require a publication prior to graduation. But, I am finding myself significantly hindered in my -academic- job search by not having any sort of publications prior to graduation. So for others who are reading this answer: while publishing might not always be necessary to get a degree you might want to keep in mind what you want to do after you graduate.
To answer your question title: I'm sure if someone was interested they could find one that doesn't, however...
The purpose of a PhD (in an optimistic sense) is to help further your chosen field. It doesn't matter how smart you are if you don't further anything, and the primary mode of this is a research paper.
Skolkovo seems to be especially strict about this, since most programs I see require only one: your dissertation. This is technically a published research paper, even if it is given a big, final boss-sounding name. Oftentimes, one will be part of multiple research papers anyways due to other requirements such as being a research assistant or just being in a research group, however there it isn't the publication itself that is important.
The requirements listed on that website are very soft. 2 can be conference proceedings and the remaining one must be peer reviewed but not necessarily first author. If you hang around a lab long enough you can accomplish that.
Institutions differ in specific requirements but nearly all, at least in the US, require some type of publication. But, that publication can be in-house, I.e., your dissertation is probably indexed by Scopus.
These are actually not difficult requirements as stated. In comparison, my last university required grad students have 3 peer reviewed articles to even be considered for PhD candidacy. That’s a high bar.
The requirement, like that of Skoltech, will vary from region to region, from country to country, and from institution to institution.
However, to redirect your 'question', it might be best to look at the concept of doctorate within the context of higher degrees.
Essentially the requirement, as stipulated at the bottom of the link, is basically a PhD by Publication - Prospective Route.
This is common in parts of Europe like the Scandinavian Countries and in Australia. In the UK, it would often be referred to 'Alternate Thesis Format' to distinguish it from the PhD by Publication - Retrospective Route (typically referred to as PhD by Published Work, or PhD by Prior Publication).
The requirement for sole-author or first author is to gauge the degree of individual effort as required for doctorate; the independent contribution to knowledge.
PS: I'm busy with a manuscript on this. I should share a preprint later. You can also read academic papers and blogs on PhD by Publication.
Publication requirements and Conference participation for PhD students
PhD students are expected to produce an original PhD thesis (check previous PhD Thesis Defenses), publish the results of their independent research in peer-reviewed academic journals, and present them at reputable international conferences.
Three peer-reviewed publications, of which two appear in WoS/Scopus indexed journals/conference proceedings.
At least two publications in:
(i) Q1/Q2 journals in the corresponding fields or journals with impact factor (IF) > 2,
(ii) conference proceedings having a CORE rating of A/A*;
One (out of two) first/shared first/corresponding author publication.