In Scotland the territorial designation is legally part of the surname (for those who use it). The Scottish law and custom is more analogous to France than to England. There is extensive literature on this subject because it is so confusing for Americans. See, for example, http://www.scotarmigers.net/pdfs/Information%20Leaflet%20No.%2020%2...
The territorial designation properly belongs to the head of the family, his wife (if she uses a married name), the eldest son, and all daughters. Younger sons get the plain surname without a territorial designation, unless they have land of their own.
The practice of duplicating the surname (Udny of Udny, or Udny of that Ilk) arose because some families take their surname from an estate, while in other families the "estate" is the clan.
Udny of Udny and Udny of that Ilk mean the same thing. Which form a family uses depends on its own preference, except that daughters will always use the form Udny of Udny rather than Udny of that Ilk. Not only do different families follow different patterns, but the same family can switch back and forth over the generations. This is one place where we have to actually do the research, not just make it up.
The territorial designation does not come from noble titles, just from landed estates, because the whole point is that it is a name, so it survives attainder.
On Geni, the whole surname (Udny of Udny or Udny of that Ilk) should go in the surname field of Scottish families, where appropriate. In England, the same kind of territorial designation would be properly set off with a comma (that is, be put in the suffix field).