The various centers coordinate on setting up the invests; I'm not sure it's any one group that assigns them, but it's the Navy's website that shows them most prominently. The NHC has the final say on actually classifying a tropical cyclone in the Atlantic and E. Pacific. In the NW Pacific, it's the JTWC. The JTWC does their own advisories for the other basins that way (incl. the Indian Ocean) as well, though there are other agencies (e.g. Australia's Bureau of Meteorology) that issue their own advisories as well.

Just a note to add that the CPHC (Honolulu) has these same responsibilities for the Central Pacific basin.