The main drawback of exaggerating within group similarity is that it can lead to excessive nationalism. Nationalism is an ideology that emphasizes solidarity within the group and (often) antipathy towards other groups.
When governments or other groups focus on what they say is the great similarity and solidarity among people of their group it can lead to even more intense dislike of other groups. The most terrible example of this, of course, was the extreme nationalism espoused by the Nazis before and during World War II. That group emphasized very strongly that all Germans had something very important in common. They argued that Germans were superior to other races. They particularly contrasted Germans to Jews, who they saw was inferior and destructive.
When the Nazis emphasized this, they came to create a situation in which Germans looked down on Jews to such a great extent that they were willing to allow the Jews to be killed and even to participate in the genocide. This is the sort of problem that can arise if within group similarities are exaggerated.