I wouldn't go that far; the [geog.] marking may actually be a helpful hint to the difference in context. (If I understand the LEO marking system correctly.)
This seems to be a change that the country Ukraine has been trying to persuade the English-speaking world to adopt since it gained independence. The politically correct, and polite, response is to abide by their wishes when referring to the official name of the country, which is now Ukraine alone (which I assume would be marked [pol.]?).
But in nonofficial contexts, some English speakers might still indeed say 'the Ukraine,' especially for the geographic area (the region) as opposed to the political entity (the nation). Some traditionalists may even still say it for the country, because it just sounds wrong to us without it, unidiomatic.
It's simply a term that's still in flux. One of the two options will probably win out eventually, but for now, both options still coexist in slightly different contexts.
Comparable cases are Bombay/Mumbai, Peking/Beijing, the Sudan/Sudan; or more recently, Calcutta/Kolkata, Ivory Coast/Côte d'Ivoire, etc. Especially with this latter group, the trend toward using the foreign word may not be as clear-cut as the proponents of the new terms might wish. The unfamiliar terms still sound very strange to many people, and there has been some growing resistance to the effort to impose foreign words on English. No one is trying to make us turn Munich back into München (we hope); why should we use a foreign word when we've already got one?
On the 'the' issue, I think actually it will eventually be only Ukraine and only Sudan, because the difference is minor enough that we can eventually get used to it. I'm just not sure we've reached that point yet; maybe in another 10 or 20 years.