You can have multiple emblems equipped and the same goes with flags, it can show you that you have multiple equipped but you really only have one equipped. Hopefully that answer that.

This happens when you use loadouts. If you use multiple flags and emblems across saved presets then when you load one of them it will load the flags and emblems from every single one you have saved.

As stated above of course, only one will display. It is as easy as changing your emblem on and off the desired one (and same for flag) when you encounter it.