as far as I know, your first capital ship - the one you get for free - is designated flagship. it's the only one and if destroyed, you cannot get another one. and there is no bonus or any other gameplay impact, it's purely aesthetical. at least I'm pretty sure of it.

The "Flagship" designation is only given to the highest level capital ship in your fleet, regardless of when it was built (1st, 2nd, or 10th). No capital will have this designation if there are two (or more) capitals with the same level and they are all the highest leveled capitals.

In the Empire Tree, if my memory is accurate, the caps are sorted by level, then order of construction. If you had all level 7 capitals, for instance, they would be sorted in order they were built from oldest to newest when they're all in the same gravity well.

I don't get it though, I remember once that I had several flag ships

What are you talking about? Whatever ship you have selected during "create fleet" will be the flagship.

This has been complained about numerous times...(mostly by me) and by others.