It is also reasonable to argue that they and we are subspecies of the same species not rather than separate species.

Of course we are not separate species since successful interbreeding did happen (when the populations interacted).

If you look closely at it, the concept of "species" is impossible to define exactly.

That's fine in general. Many words are like that. But it makes it flawed as a scientific concept.

Donkeys and horses are not the same species, but they interbreed. Their offspring are called mules. Although mules rarely have offspring, so perhaps not a great example.

There is supposedly neanderthal DNA in some modern humans, implying that offspring were viable. Breeding resulting in viable offspring is one of the only consistent definitions of what a species even is.

We could make that the definition, but we'd doing a lot of redefinition: coyotes and wolves would become the same species, as would lions and jaguars. Fertility issues tend to increase with genetic distance but aren't guaranteed; for example, mules are usually but not always sterile.

What about the 2nd generation of mule, then 3rd? If the probability of viability keeps dropping generation after generation, eventually it will delete itself?

It doesn't work like that. Backcrossing it with one of the parent species (assuming fertility in the first place) as you'd expect tends to increase the likelihood of fertile offspring in proportion to the number of generations. And that's exactly what you'd expect in any hybridization event. And anyway, mules are actually pretty special in that horses and donkeys are actually fairly distant relatives (diverged 4 million years ago) and have different numbers of chromosomes. All members of homo (supposed to have emerged all more recently than 3 million years ago) could probably interbreed and the ones that had the opportunity probably did.

It's really unfortunate that schools tend to simplify the definition of species in this way, because it's just not really not meaningfully true at all. We could "make" it true by actually defining species this way (at least for animals) but it'd radically transform our taxonomies.

That's not the definition of species. Many species can interbreed, sometimes even species that are pretty distantly related.

I have never heard anyone refer to Neanderthal as a human unless they are talking in a "those are cavemen, early humans" way that's wrong. Where is this coming from? Is it a non-English world thing?

Generally Neanderthals are pointed out as an exception to cross species fertility since... humans have some Neanderthal DNA.

> Generally Neanderthals are pointed out as an exception to cross species fertility since...

There's no such rule and Neanderthals are not notable as an exception. Fertility is just a very rough proxy for genetic distance, which is correlated to our arbitrary "species" buckets but by no means a real line or hard rule. Many, many reasonably closely related species can interbreed, like jaguars and lions. Most of homo that had the opportunity could probably interbreed.

Wherever it’s coming from, it’s not based on any particular language. Museums in English-speaking countries use the term.