The word "Kilimanjaro" is used in the song only for rhyming and it's catchy. The composer of this song is A.R.Rahman. He prefers catchy words in his songs and also asks lyricists to use a simple and catchy words in the songs. We can find many examples from his famous songs. Some of them are
Ballelakka from Shivaji The Boss (We can find a more catchy pair of words in the song).
Chikubuku Chikubuku from Gentle Man etc.,
The unclear word which was used in the beginning of the song Kadhal Anakkul (Neelo Valapu in Telugu) is not intended to be in the song but A.R Rehman liked it when the singer of the song Hariharan used to clear his voice before singing the song.
Humma Humma song from Bombay and its remix is used in OK Jaanu.
Similarly, we can find more examples if we dig deep.
Now, your question can be understood in two ways.
Why is the song is shot at Machupichu?
Why isn't this song shot at Mt.Kilimanjaro?
1. Why is the song is shot at Machupichu?
The director of the movie is Shankar. He likes to show new locations to his audience. He shot a song from one of his movie Jeans at old seven wonders of the world. That song was a sensation at the time of the release as well as the movie. Endhiran was the first Indian movie to shoot a complete song at Machupichu. That too after direct interference from Government of India.
2. Why isn't this song shot in Kilimanjaro?
Kilimanjaro is the tallest mountain in Africa. It rises approximately 4,900 metres (16,100 ft) from its base to 5,895 metres (19,341 ft) above sea level. It is next to impossible to climb a mountain without proper training. Mountaineers have to pass many medical tests to climb that mountain. At that level, breathing is not easy. Filming an entire song is impossible as well. No wise director would put their actors' lie in Jeopardy for a film. Moreover, the song was just a duet between hero and heroine. Kilimanjaro has nothing to do with the movie's plot. So, no one would risk shooting a 5-minute song on a dangerous mountain.
See Is Climbing Kilimanjaro Safe?
So we can conclude that the word Kilimanjaro in the song is only for rhyming purposes. It has nothing to do with Kilimanjaro mountain or with the plot of the movie.
By your logic, we can also ask another question.
Why isn't the song shot at Mohenjo daro as that word appears in the second line?