I read this yesterday and wrote a comment on the article at Medium but I think there's more nuance than the 400-character comment limit there allows (although I think such a small limit on comments is a good idea for that site's format).
It's true that by the strict definition, nothing is rare anymore if it's on the internet. But there's still a difference between something everyone always had access to, and something that was a physical rarity in "olden times" pre-internet. We're not quite at the point where absolutely everything is immediately accessible.
All of the examples he used were things that were once rare. Truly, there is irony in labeling something "rare" on youtube, where everyone in the world has access to it. But in practical terms, it still does mean that today they are still rarely seen - even if the acoustic version of Sweet Child O' Mine has a couple million views on youtube, that pales in comparison to the billions of people who have probably heard the original song by now.
But the real issue here seems to be the concept of being "media cool", the thinking that knowing about obscure stuff has value. Setting aside the comic-book-guy types, doesn't it though? This is a big part of boingboing... as an obvious example, the editors here sometimes bring us interesting, obscure musical recordings or performances that have little value to most people (recently, the Velvet Underground concert with bad sound) but which we (or some of us anyway) find interesting.
So "rare" has a deeper meaning in this context - it's something that a) isn't widely known/heard/seen, for a start. Sometimes, it's something that would be interesting to a lot of people... although I'm not sure any of the article author's examples count. Not even obscure Beatles recordings, the most interesting of which are commercially available for those who are interested anyway. So, important to this discussion is that often it's also the case that b) these are things that don't really have wide appeal. Which is the same as ever; band b-sides and rarities being a good example - the stuff with the broadest appeal gets released and promoted; the extras may have very high appeal but only to a select group of hard-core fans.
Finally, with that in mind, rarity in its original meaning is still relevant on the internet. As I mentioned at the beginning, we are not quite at a point where everything ever created is available to us. There are still things - music, film, TV, everything - that don't even qualify as rare in the original sense, but which are still impossible to find on the internet. As someone whose tastes often run to the obscure (as is the case with many boingboing readers I'm sure), I quite frequently am stumped trying to find something I am interested in, either through legitimate means or otherwise. And b-sides and EPs, even from really well-known bands, are often out of print and can still be impossible to find anywhere even to download ripped mp3s of. That still counts as being "rare", surely.
And even when we do get to the point where everything ever is accessible immediately, things will still be rare by the deeper definition of "rarely seen and/or of rare interest", because there's just too much stuff for everyone to be familiar with everything.