It was known before that nucleic acids can self-replicate, and that RNA could also do some protein-like functions on top of that. This was proposed in 1962 as a starting point for the DNA-protein system as we now know it, which seemed to solve a problem we had had before: protein couldn't do DNA's job and DNA couldn't do protein's jobs, so nobody knew how to get an interdependent cycle like that started because neither could get going without the other already there to help it. With RNA known to be able to do some of both kinds of functions, it could get started all by itself and then later hand off some things to other molecules that could handle them more efficiently/reliably.
It was also known for the last few decades that lipids in an organic acid solution forms microscopic sheets which could spontaneously curve back on themselves and enclose a volume inside, which, through entirely thermodynamic processes without a single catalyst, would then spontaneously grow until they were too big to remain intact, and then split without breaking & spilling their contents, just like a cellular membrane except with no genome or organelles inside it. (When they touch each other, they also tend to merge, which, when one was much bigger to start with, is pretty much the same thing as the big one consuming the little one.)
The latter is what RecoveringYuppy's first link describes. But it's about lipids forming something like our own cells' lipid membranes, not anything that addresses the usual meaning of questions of "self-replication", which is about the nucleic acids and proteins inside such a membrane. You could call these membrane-bubbles "self-replicating" in a sense, because of the growing & splitting, but there is a fundamental difference from what we normally mean when we talk about "self-replication". Aside from the fact that it's a lipid not a protein or nucleic acid, it's also not very specific. One bubble, or one molecule within such a bubble, is pretty much interchangible with any other, just like with crystals, which were mentioned above. In nucleic acids & proteins, just generating more random nucleic acids & proteins like that would never count because their components are not interchangible. You need the same particular kinds of pieces in the same places. GATTACA is not the same thing as TAGCTGA, so "self-replication" means GATTACA creating GATTACA, or TAGCTGA creating TAGCTGA, but not just any seven units creating any other random seven units. That's what always kept protein from being considered "self-replicating" all along. The same conditions that created one generation of protein molecules would also create another, but they wouldn't be duplicates of the previous generation. Only nucleic acids have been known to do that until pretty recently.
RecoveringYuppy's second link describes a bit of research from 2017 which they found true self-replication (not just the equivalent of crystal growth or membrane-bubble growth), and construction of a larger (but still microscopic) object from the copies, in a protein. But it was an artificially engineered protein assembling its copies into a sphere under controlled conditions, not an indication of whether or not any naturally-occurring proteins could replicate or assemble naturally-occurring or physiologically-useful shapes in a natural setting.
From the way the link in our original post here uses the word "replication", I'm not sure whether they mean point-for-point distinction between components, as in DNA, or just "proteins from proteins" regardless of whether any two molecules are the same or not, which would only qualify under the looser definition that also allows lipid membrane bubbles and crystals. Either way, though, the researchers found two separate things: that it can happen in coacervate droplets, which are a much more natural formation we've known about for a hundred years but not known could do this; and that the proteins in this case were also collecting nucleic acids, which is important because most proteins don't do that but the ones in a nucleus do, although the article is vague about the exact nature of this interaction. (It's also vague about what makes these coacervates able to self-replicate if other coacervates don't.)
So... first known self-replicating molecule? No, that was ages ago. (But the article doesn't claim it was.)
First self-replicating protein? No, that looks like it was about 4 years ago. (But the article doesn't claim it was.)
First self-replicating protein in a scenario that's more like what could be or could have been found in nature, and also manipulating nucleic acids while it's at it? Yes, apparently so.
What it means for the origin of life: It might be a sign of a possible alternative to RNA-first which could have happened in nature: either protein-first or a combination of proteins and nucleic acids already interacting with each other from the start.