> many animals can distinguish independently small numbers, like 3 or 5

This is called subitizing. It's distinct from counting. We can see the difference in humans with Simultanagnosia, who are unable to count beyond the subitizing range. Subitizing is categorizing the scale of a small gestalt group.

The only thing I've ever seen where an animal appeared to demonstrate counting (up to 3) without training was in rhesus monkeys (maybe also chimpanzees?), but even that experiment could be explained through temporal gestalt. (It's the only reason I know of for them to not have been able to go higher than 3 in that experiment in the context of many other things that they can do.)