As I understand it, the tradition of wearing clan tartans in military service started with the raising of certain regiments under a particular clan chief or title holder, who would then choose to have 'his' tartan as the uniform of the regiment. Over time, those clan tartans become official uniforms of the regiments as they were brought more formally into the British Army. And in many ways this stayed true to the original concept of clan membership. I think people sometimes forget that clans weren't just people related by blood or having a particular surname. They were made up of families and individuals who swore fealty to the chief and fought for him, or lived on his lands. Of course, the clan system really died before clan tartans came into vogue. But when it comes to serving in a Highland regiment, the wearing of the tartan associated with that regiment (which originated from the clan tartan) is a nod to the old traditions of serving a clan, and the new tradition of being identified with it by wearing its tartan. Only in this case, it's a military unit and not a clan.

The same tradition morphed into pipe bands, which of course had military origins. But also probably originated with the chieftain's piper having the honour of wearing the chief's tartan and flying his personal arms, etc. The fact that civilian pipe bands are not tied to a particular clan or military unit doesn't matter. They are just carrying on the tradition. If they choose to wear a particular clan's tartan, it is accepted in that context. It's a uniform, and the traditions of Highland uniforms revolve around tartan wearing.

That's not at all the same for individuals.