I think you're confusing the flagship designation with the fleet leader.
A ship that has been assigned as the leader of a fleet will have an icon over it and when you click on that icon, the entire fleet is selected. All that's required is for you to select one or more ships and create a fleet with those ships. The sub-selected ship becomes the fleet leader.
The flagship designation, by contrast, is EXCLUSIVE to capital ships and is assigned only when you have one capital ship that is the highest level AND is the only capital ship at that level. Once you have two level 10 capital ships, none of your capitals will have "Flagship" before their class on their infocard until one of those level 10s is lost.