It's true! I can't believe it, either! But the evidence is there...
Wealthy Private Organization: In 00, it's Celestial Being. In Thunderbirds, it's International Rescue
Powerful machines utilizing secret technology: 00=Gundams, Thunderbirds=the Thunderbirds
Heiress lady with a manservant who serves in a support manner: 00, it's Wang Liu Mei and Hong Long, Thunderbirds, it's Lady Penelope and Parker.
Organization attempts to better the world in a vigilante manner: Celestial Being is a tad more forceful than International Rescue, but IR still took out its number of bad guys.
Value of secrecy: Both organizations value secrecy, however, IR destroys any images of the Thunderbirds or pilots, whereas Celestial Being pilots just remain masked.
Mission planning from supergeniuses: Thunderbirds' head brain is Brains. (Actual name of the character) In 00 it's Sumeragi.
Private Island: Both organizations of both series use a private island as a base. Both also have outposts in space.