Your tag is based on reputation if you are crewless. If you join a faction as one org, then change rep, you will stay within that faction. However if/when you left that faction, you would then display as a different org.
If you do the storyline (intro) missions for each org, i.e. the Tyndall, Agent Gray, and Merv starter missions, that'll affect your rep. You CAN do them, it just would have adverse effect on your rep. (Really, the starter missions aren't that story related, they just ease newbies into playing MXO. I'm not sure if by story missions you mean the starter missions or archives.)
Archive missions don't affect your rep at all. You can do all the archive missions over and over, for all orgs, as many times as you like.
