An organism will always be unique from the other even if they do have the same amino acid sequence.
Each amino acid can be coded by three DNA bases (A/T/C/G), a codon. These codons codes for a specific amino acid, but a single amino acid could have diferrent DNA base sequence. For example, Phenylalanine could have a codon of UUU or UUC.
I hope this helps. Feel free to ask me if you find this confusing. :)