They may well use sneakers over trainers there. Its a commonly enough understood term in English speaking countries round the world though
And the song was written back in *thinks* 2004 and performed in the UK and elsewhere way before it made it to the US in 2005 on the HBO special, so no I don't think it was written for a US audience.
I'm more inclined to think it was just a better word for the flow of the song
Hope that helps