Each of us were to pick one that had meaning to us..., to pick a mon and carry on the tradition.

I can get behind that sentiment, as it rationally fits within the sense of modern and international iaido study. Just like the kamon, a particular ryu used to be a closely guarded local tradition. Today, people who have no familial or geographical connection to the koryu still study them. If we didn't, the koryu would be gone. Nobody has legal claim to them: the bushi class is no more. The few who have "valid" claim to them geographically or by family lineage are less and less likely to also have the means to uphold them with a dedicated lifestyle.

The other example was Scot tartans. If a Chinese guy respectfully wore a kilt to a Scotland sporting event, I think it would probably get him a free mug of beer, regardless of the family indicated by the weave. Of course, if the Chinese guy was otherwise acting like a jerk, there might be some who took offense and the cultural garb might exacerbate it.