What is the only way that elements can be separated

Well if you mean 2 elements together then by chemical mean but if its just one element there is no way of separating it

If you separate an element into singular atoms, it will no longer be 'that element'.

It takes an unimaginable amount of energy to even try to separate most elements to atomic forms. Even if we did, the atoms would be too quick to hold in their singular state, as they would bond with surrounding atoms or re-bond to the existing ones.