Not a scientist, but I think the BFD is that it's 30k years old from a completely extinct branch of animals (unlike mammoths which are closely related to modern elephants) and the thing they found was a frozen tundra mummy not a fossil so it has soft tissue and stuff unlike other extinct animals we typically observe via fossils like dinosaurs

That's an understandable take and the use of the word analogue is the key issue. It also left me stumped for a while, because as you have already pointed out, there are plenty of modern day analogues to Homotherium...

...but that depends on what counts as an analogue in this particular context. Biologically speaking, the word can be used to fit a broad range of criteria. So you could say their modern day analogues are lynxes or snow leopards, and fair enough, that would be a good enough use of the word because these animals do share a lot in common, physically and in their ecological roles too. Large catlike mammal that hunts down larger herbivore mammals in a tundra environment.

But Homotherium had some very specific traits that have no modern day analogues. The large canine teeth is the most obvious. Those large teeth also meant a specialized hunting method and technique for which we have nothing alive today to base it off of. They also had a different body build, with shorter rear limbs, so now your cat looks a bit more like a bear or a hyena in its stance and gait. And if I'm understanding what I'm reading correctly, they also had cardiovascular adaptations for endurance running, and their claws and paws were not as retractable and supple as that of cats.

So yes they were like cats and you can point at living analogues for a lot of these adaptations ( bears, hyenas, cats, any mammal with good cardio, etc) but when you put all that together and add the teeth and the behavior modifications those teeth imply then you have, as a whole, an animal with no current living analogues. Yes, it can sound pedantic but that's science for you and I think it's important to remark that the quote is taken directly from the paper published. The journalists loved the buzz emanating from the word "analogue" so much they kept it in the non scientific publications, they didn't paraphrase, and they didn't bother explaining exactly what it meant because, well, that's precisely why they chose to keep the quote.

I think this image on the Felidae wiki sums it up pretty well: https://en.wikipedia.org/wiki/Felidae#Phylogeny

Note that everything in this graph is extinct, except the 2 circled subfamilies at the bottom.

EDIT: I basically know nothing about biology and paleontology and am just an amateur wiki-binger, but it seems that the half-whale was further away philogenetically from any live mammals.

I'm guessing that the fact that this is not just some bones but a very well-preserved mummy is what makes this find special.