Walt Whitman repeats the word singing in his poem. Is there a hidden meaning behind this repetition?

This is a good question. There isn't really a "hidden" meaning in the sense of something Whitman deliberately made difficult or opaque. On the other hand, "Song of Myself" was first published in 1855, and there are assumptions Whitman makes about how the term would be understood by a nineteenth-century audience which may not hold for a twenty-first century one. Thus some sense of the cultural background will be helpful.
First, education in nineteenth-century Europe, Britain, and North America was based on the Greek and Roman classics. If you attended school in this period, you would have started learning Latin in primary school and then added ancient Greek in middle or high school. You would have read little or no literature in your own modern language. Ancient epics were sung rather than read. The standard works any schoolchild knew tended to begin with a mention of singing. Virgil's Aeneid, for example, begins "I sing of the arms and the man," and Homer's Iliad starts with the lines "Sing, Goddess, of the rage of Achilles, son of Peleus." Thus when Whitman composes what he considers an American epic, he follows epic tradition in starting with a mention of song: 

I CELEBRATE myself, and sing myself ...

What is unusual is not the singing, but the fact that he is singing about himself rather than a historical hero. 
Additionally, in the minds of Whitman and his compatriots, song tended to be connected to a popular or folk tradition, as opposed to written literature, which was though to spring from a more modern and elitist tradition. Thus by presenting himself as singing, Whitman is contextualizing himself as part of a popular folk tradition allied with the common people as opposed to a more literary or academic tradition.