Every word of these two paragraphs remains true when you replace replace "animal" with "toddler" and "human" with "mother". Have you just debunked that mothers can teach their toddlers how to speak?
There is a big, scientifically testable, difference between human toddler, and non-human animals.
It is pretty easy to show that toddlers understand the meaning of words at an early age. Very quickly they use words to express new ideas, and they can understand sentences that they have never heard before based on their understanding of the words. Toddlers will show language skills consistently... not over thousands of combinations (poured over by researchers to find meaning in any of them... toddler will quickly show understanding in every utterance.
The important part... this ability in human toddlers is science... it is reproducible and measurable even when tested by independent researchers. This is not true in any of the claims of animal language.
The many claims that animals are able to do these things... putting words together to make new meaning, or understanding sentences they have never encountered are anecdotal. If an animal could consistently put words together to express new meaning (rather then being selected out of many combinations of symbols, or selected by human feedback to get a desired reaction from humans), then it would be something.
That humans (including toddlers) use language is scientifically testable and confirmed. Claims that animals use language has never met scientific muster from independent researchers.