Nobody ever said that all cells share 100% of DNA in the sense you're implying. It's obvious that DNA undergoes dynamic processes, some of which will result in lasting differences. The accumulation of errors is the basic mechanism for cancer, which has been recognised for decades.
But it was previously thought that such differences were detrimental to function (and health). There were some mechanisms for modifying cells' DNA, collectively known as "epigenetics", but these were reversible, at least in principle.
This adds another dimension of complexity, which biology already had plenty of. My favourite is the double- or triple-coding DNA: because DNA is read in triplets, there are three possible reading frames. Some organisms have evolved DNA that produces entirely different, but independently useful proteins on two or even all three reading frames of the same DNA. Talk about efficiency...