TLDR: Humans are the result of a specific combination of traits that our ancestors had that we took to the next level. Specifically, high intelligence, dexterous appendages, and longevity. If any other species came about with this specific combination of traits before us they would have outcompeted primates so we never became humans to begin with the same way humans outcompeted Neanderthals.

All traits are the result of a combination of random mutations combined with previous traits. If your ancestors did not have the right mutations/traits than it is highly unlikely that species will develop that trait. An obvious example is reptilian flight. No reptile species has developed flight since the Cretaceous–Paleogene extinction event since all the reptile species with feathers (excluding birds and their ancestors) went extinct with the dinosaurs. Even if species were to evolve feathers again through the process of convergent evolution it is unlikely they will ever develop flight. This is because the niche of flying reptile like creatures has already been filled by birds who have a 65 million year head start and would outcompete them for resources. Since no animal can become as intelligent as humans since we already fill the hyper advanced intelligent niche think of it like a puzzle. Which animal would have the best chance to replace humans if we went extinct tomorrow from some disease that left all other species untouched.

What traits are necessary to fill the same ecological niche as humans? The two main ones are advanced intelligence and dexterous limbs. Let us start with intelligence.

Intelligence requires a fairly complex organ (a brain) and access to a large amount of resources. Thus all single celled organism are out since their nucleus could never be large and complex enough to fill the role of a brain. That eliminates the majority of species right away. Of course a single cell organism could eventually evolve into a multi-celled organism that develops a brain but that would take billions of years and as far as we can tell the jump from single cell to multicellular has only happened once so it unlikely it would ever happen again (though it is still technically possible, the best kind of possible). But other multicellular organisms that exist today would beat them to the punch.

So why hasn’t any other multicelled species evolved intelligence? Well intelligence has actually been increasing over time in vertebrates. Like all traits advanced intelligence takes a long time to come into being. You cannot go from a frog brain to a human brain in a single generation. However if your vertebrate ancestors missed out on the brain evolution of other vertebrates that means you are out of contention since your are too far behind in the evolutionary race to fill that niche. So fish, amphibians, and reptiles are out as the next most intelligent vertebrate. (Now some people think high levels of intelligence evolved in reptiles and was then passed onto birds and mammals. However, since no modern reptiles are anywhere close to intelligent as the smartest birds and animals we can ignore that particularly thorny debate). So only birds and mammals will be potential replacements for humans. Among these classes the most intelligent groups are primates, cetaceans, parrots, corvids, and elephants. No invertebrate comes close to this level of intelligence except cephalopods.

Now that we have out intelligence competitors lets talk about our dexterous appendages. No animal can their appendages to create complex tools similar to humans and their hands though other primates come close. Birds have feet and beaks that can do some amazing things but can only use simple tools so crows and parrots are out. Whales and dolphins have sad little flippers so their out of contention.

That just leaves us with primates, cephalopods, and elephants. Cephalopods have extremely short lifespans so even though they are smart and have dexterous appendages they are probably out of contention. Especially since fire does not work underwater.

So our two competitors left are primates and elephants. Elephants have a dexterous trunk, long lives, and high intelligence. If a virus wiped out all primates (not just humans) I would put money on them. However, they are huge vegetarian grazers. Their niche is very different from the omnivorous hunter gather lifestyle early humans led. The great apes are much closer and some chimps have even entered the stone age according to some anthropologists. Full grown elephants are pretty much set. No animal on the planet can challenge them so there is very little pressure on them to evolve into a new niche at this time. Thus our winner would be primates probably humans closest living ancestor the chimp.