Just call yourself a "group" instead of a "clan" and it should be fine and dandy. I see many groups of friends who play together, or at least I used to. Factions are classed as Clans and viceversa here, as Factions were known as clans in the past. If the two suddenly differentiated, there would be confusion. Such a ruling by the admins is probably just to avoid this confusion, rather than to in some way reinforce "buddy buddy" relationships which dont exist.
Factions and clans must have the same IDs. It would be pretty ludicrous to see a [LN] with a trader ID, as militaries dont trade. Equally as stupid as a [OPG] tagged player with a smuggler ID. In this case, the OPG have a seperate wing called the [OTC] for that. Even if it wasn't so ludicrous in whatever circumstances can be dreampt up here, it would certainly cause confusion for those not within such a clan or faction.
As I understand it, groups like this dont subscribe to that. You can make a group that looks official and is official, minus the faction status post. You can even have feedback. The system as it stands fully facilitates this kind of play, but such a motley crew would have no need to be "official".