It appears that you are treating "is" as an equality operator. There are contexts where "is" denotes equality ("one plus one is two"), but in other cases where it denotes something else, such as subset ("cats are mammals") or attribute ("cats are furry"). If we replace "cats being born in the world" with "A", "natural" with "B", and "rabbits being born in the world" with "B", then your argument is "A is B, C is not A, therefore C is not B." If "is" is denoting equality, then this would be a valid argument. But the first and last "is" are denoting attribute. The argument "Cats are furry, dogs are not cats, therefore dogs are not furry" would be a shorter version of this fallacy. Or "five is prime, seven is not five, therefore seven is not prime".
This can be seen as an equivocation fallacy (using "is" in different sense), denying the antecedent (we have the true statement "if C were A, then C would be B", and you're denying the antecedent "C is A" to negate the conclusion), false dichotomy (the argument boils down to claiming that everything is either A or not B).