Considering Renly’s Rainbow Guard, it seems to me that people like Caron are able to keep their titles so it has no impact on their property. Obviously the group were more like an association than an organization, and didn’t have the time to get off the ground, but are there any comparable real world organizations?

Well, yes. Historical royal knightly orders generally didn’t require people to take vows of poverty and chastity, in contrast to the historical monastic knightly orders. Given that important noblemen are pretty insistent about being landowners and siring an heir, it puts something of a damper on recruitment, since the religious folks are alll going to join the Templars or Hospitallers instead. 

Mostly, royal knightly orders were about symbolic politics, showing off how awesome and special the king was, and allowing the king to draw powerful and important nobility close to him by giving away shiny medals instead of land or government offices, and everyone knows that nobility love the opportunity to dress up in really stupid costumes and throw crazy parties. Viz:

That’s not to say there weren’t eligibility restrictions or requirements on members. For example, you had to be already a knight to be a member of the Order of the Garter and you had to be elected by the existing membership, and if you were one of the “poor knights” (impoverished veterans made additional members of the Order outside of the 24 elected members) you were required to pray daily for the elected members in exchange for your pension. Likewise, for the longest time, you had to be a Catholic to be a member of the Order of the Golden Fleece, the royal order of the Hapsburgs, which was the tradition from 1430 to 1812, when Wellington was made a member because of the whole saving Spain from Napoleon thing.