Because not all things "age" at the same rate. (1 month old milk is old, a 1 month old can of tomato paste is far from old)
Lodash (#1 in this article) is just as old or older than Moment, however I wouldn't call lodash "aging" simply because it has adapted to the new paradigms in the javascript ecosystem and works very VERY well with them (to the point that I'd argue lodash is a pretty good example of how to work with these tools).
Moment still does a great job at what it does, and it's in no way getting "worse" over time (quite the opposite!), however because of some choices made during it's development, it can't easily take advantage of newer developments in the javascript ecosystem.
JS has changed significantly, and is still changing. I'd argue that JS became a "real" programming language over the last 10 years, and because of that a lot of things created in that time aged much quicker than they should have.
Like I (briefly) pointed out in my comment, things like tree shaking, module bundling, code splitting, minification, and the tooling and ecosystem around them have changed drastically in the last 5 years. Now moment will still work great with those things, but it doesn't take advantage of them. And that means that alternatives end up getting a leg up because they do take advantage of them.
Something like `date-fns` is designed with tree-shaking in mind, so if I only want to use the part of it that "humanizes" a time range, i'll only get that code. If I want to provide multiple locales for multiple users, but only have each user download their own locale at runtime, I can do that easily. Not to mention that browser support has changed in the last few years, so most people no longer need to support IE 7, 8, 9, etc... And that means that the code that moment has built up to support those browsers isn't needed on newer ones, but again it's difficult or impossible to build it without that to save size.
To continue the analogy I used at the start of my comment, I'd say moment is week old milk. It's not bad yet, but I wouldn't buy it from the store like that in most cases. Moment is great, I do use it in 2 projects at work because it was the best tool for the job for a long time, and it still works fantastically. However if I'm starting a new project now, i'm most likely going to reach for something else.
Not because it's "newer" or because it's the hot thing right now, but because in all honesty the API of a date management library isn't all that complicated, and switching isn't a large cost. And when those other libraries can provide benefits that moment can't, i don't have much of a reason to stay with these "older" libraries.