"Animals", as you put it, is a construct. There are no real species, because every discrete animal is the same species as its parents, as wel as a species onto itself, since it has a lot of DNA in common with its parents, but the combination is unique.
The only reason for the taxonomy is to make it easier to talk about certain groups of animals that, at a certain point in time, were/are similar enough to practically belong to the same species, so that we don't have to name every single animal individually.
I disagree. A "species" is defined as those animals that can interbreed with each other to produce fertile offspring. Well, that's one of the definitions, and it is fuzzy around the edges.
But what all members of a species - say, humans - have in common is not that they have exactly the same genome, but that their chromosomes do have the same layout (*): how many chromosomes there are, and where the genes are located in the chromosomes. What makes the difference between individual humans is that they have different combinations of alleles on the loci for those genes. One person has two alleles for brown hair and two alleles for brown eyes, and another one brown hair and one blonde hair allele and two alleles for blue eyes, etc. etc. With the number of genes which have different possible alleles the number of combinations is virtually infinite; "possible" meaning an allele leading to a viable organism.
(*) Barring all kind of copying mishaps, such as trisomy of chromosome #21 (Down syndrome), or excessive replication of a part of a gene (as in Huntingdon's disease).
I think the OP wondered whether maybe Asian elephants are descended from woolly mammoths, in the same way that birds are descended from certain dinosaurs.
I'm not sure what the OP meant, maybe he'll clarify it.
But in short, the cloning effort will take an Indian elephant's egg, and replace the nuclear DNA with nuclear DNA from a mammoth. And because the DNA of both is very close, they hope that the thus developing embryo is compatible engouh with the elephant's womb that it will carry to term.
This, however, does not mean that both are the same species. Horses and donkeys are two different species and even have a different number of chromosomes, and still can interbreed (though the offspring is virtually always infertile). Lions and tigers are different species and can interbreed.