I am a fifth-year Ph.D candidate in Applied Mathematics (my general research interests include probability theory & stochastic processes & PDEs) at a research university in U.S. and I have applied for almost 80 postdoc positions (via Mathjobs.com and Interfolio) in U.S. and Europe, and up to now I just received one offer from another research university in U.S. (although I received four interviewers in total up to now, but three of them eventually rejected my applications). Actually I thought my publication record is not bad (2 published papers and 3 pre-prints at the time when I applied for postdoc positions), of course I am aware of some "super-star" applicants but I also believe that super-star applicants are relatively rare, I am just wondering how on earth the competitions for postdoc positions (not even tenure-track positions) can become so fierce? Or does such fierce competition only occur in the area of (Applied) Mathematics?
How we got here is an interesting question, but it is likely a coming together of several things. Uncertainty usually puts a damper on hiring, and it might also cause a jam up in getting employed, with many people graduating earlier still in the pool.
COVID didn't help, of course, nor did the current political situation in the US. Too many states have decided that providing tax revenue for public goods is politically impossible and so funding is poor. Universities themselves have become political targets, further affecting funding. Science itself is under political attack.
There aren't a lot of "moonshot" goals at the moment that push money into academia as there have occasionally been in the past.
I'll guess that the biggest block at the moment is just uncertainty and that may settle out (or not) once we learn how COVID will actually play out. Unrealistic views and propaganda aren't helping, and the disease isn't going to cooperate.
I, too, graduated (long ago) when the academic job market collapsed. After a huge funding boost for the original moonshot, it completely dried up overnight leaving lots of us about to earn doctorates in an empty market. Many mathematicians with doctorates had no opportunities beyond pumping gas (and, yes, people still did that for you).
Look at all your options, including some you haven't ever considered. You've been thinking about research at an R1, most likely. But there are other satisfying academic positions that can give you a good career or serve as a bridge to better times. Industrial research labs also do some interesting things in at least a few places. If you take something less than ideal and want to move up, make sure that you develop and keep collaborative relationships and do what research/writing you are able to manage. It may turn around, but don't assume that it will be soon.
I feel that around the level of post-doc, the hiring question also shifts a bit from how good you are to how well do you fit in the group. People might of course still hire a super-star candidate, should one show up, but those are rare and tend to only apply to specific places anyway.
In general, the choice will be between several roughly even candidates. And all people involved know the vagaries of the business well enough to realize that this decision does not really boil down to numbers. For example, far too often, the number of papers you have at this stage of your career depends less on how good you are than on how lucky you have been with the review process.
So people will look less at how many publications you have, or how good the respective journals are, but at their content. Does it show mathematical proficiency and some new ideas or is it just standard methods applied to a slight variation of a standard problem? Does it fit to what the rest of the group is doing? And if it doesn't fit, can it be turned into or combined with something else so that it does?
Your task as an applicant is to convince the people who are hiring of that. Sometimes this is easy. You might already work on precisely the same problems, they know you and you know them. But considering how few people tend to work in each sub-sub-sub-topic, this is rare. Much more likely is that you are only adjacent at most and as it turns out are a dozen others. So show them that there is a connection.
When I was involved in hiring a post-doc the last time, I found it really confusing how little time most people spent on customizing their application to the position. Everyone mentions the research they'd like to do, usually in continuation of what they already did before. But while this is nice, what people care about is what research you like to do with them and how the stuff you did could be combined with theirs.
I am not saying that you should write a whole new research proposal, in fact you should keep it short, not be too specific and keep it in a prominent position so that people will actually read it. Put a few paragraphs close to the beginning of the cover-letter, where for example you mention that you did X, the people you apply to recently did Y and that you think that combining the ideas on X and Y could lead to a paper on Z. There is no need for detail here, if they want to know about X, they can read your paper, they know about Y and the whole idea is to get them thinking about the possibilities of Z on their own, which gets them to invite you for an interview to discuss this some more.
I am not necessarily claiming that you did not do this, but I've noticed that far too many people are just content with just changing recipient and date in their generic cover letter. And if you say that you already applied for 80 positions, I am not sure that you spent the necessary half a day or so for each of them to skim a few of their recent works and rewrite your cover letter.