There are not two cells in the human body that have an identical DNA sequence as detailed here earlier. But not only ageing, already basic B and T cell recombination introduces variation. And there might be more: HLA micropolymorphism! A new JEM paper describes

Maintenance of HLA polymorphism reflects natural selection for enhanced protective immunity against the pathogens encountered by humans. The MHC molecules can differ from each other by either >30 amino acids or only a few amino acids (micropolymorphism). This MHC-I polymorphism is generally concentrated in the antigen-binding cleft, controlling the size and diversity of the peptide repertoire presented by each HLA molecule.

Fortunately TCR degeneracy allows even non EENLLDFVRF cells to recognize EBV, yea, yea.