Actually, Axel didn't die at that point.
You know, Axel is really good at making "fake" deaths, and he did that 2 times
in the whole KH series.
1. In Re: CoM, when you face Axel the second time, he pretended that he died,
but actually, he survived. (Evidence is shown in Riku Mode, when Axel said, 'That Sora almost had me.'
2. In KH 2, at the beginning of the game (When you are playing as Roxas.), he pretended that he died.
By what he meant by "Let's meet in our next lives..."
It means that he would meet Roxas again... But in the form of Sora.
Axel died at the point when he self-destructed himself...
The only way that Axel could die is killing himself, but not being killed by others.
Sure enough, he could be defeated, but not killed.