There are some rules of thumb for how specific a "specific" number is in practice. Say, "100" might well mean anything above 50 or below 150, whereas "10" might similarly be between 6 and 14 - but "120" would be more accurate than "100" and might fall between 115 and 125.
"Two centuries ago" is extremely vague, but no usual definition of it will cover the years from 1996 to 2268 as required by "Space Seed", or to 2284 or later as required by ST2. We need to fudge those in some other manner, such as saying that Khan personally only experienced 200-250 out of those 270 years (say, due to time dilation in his sublight spaceship), or that the characters using the "two centuries ago" expression had a reason to belittle the passage of time. Kirk would have that reason; Khan would probably want to exaggerate instead.
Now, "15 years" may appear much more precise than the above, but OTOH it is a figure typically quoted in describing the human experience, and as such is very
sensitive to belittling or exaggerating. In practice, it can be seen as an euphemism for "over 10 years", going up to 20 years if need be. In mathematical practice, anything ending in 0 or 5 can be viewed as a rounded number in our decimal system, and anything between 13 and 17 years would be rounded as 15. Khan felt he had been abandoned for this length of time, so again he might wish to exaggerate; Kirk might want to absolve him of the above sin by belittling, or then put the thing farther past him by exaggerating. In practice, something like 17 or 18 is our best bet for what the characters meant by "fifteen".