It possibly fits within the portfolio of the ITU (although I guess the UN is intergovernmental, rather than a government).
Now ICANN? They seem a bad choice, as this story demonstrates. Tolerable as long as the internet wasnt very important, but today, a UN international body seems the obvious choice.
For example, ITU also controls country code assignment for phone numbers. Does that translate to any meaningful capacity to censor? So far as I know, the only practical restriction that comes out of it is that unrecognized states don't get one assigned, but that's also generally true with TLDs.