How to change the aspect ratio of a Nikon D90 in video mode?

As the title says, I want to change it from 4:3 to 16:9.

Read the manual.

If you don't like that answer, you can brute force your way into the menu system of your camera until you find the right setting.

Taken directly from the product specs as listed by DPReview.com
Resolutions 1280 x 720 (24 fps), 640 x 424 (24 fps), 320 x 216 (24 fps)

These are all wide-screen formats, not 4:3. You have some other issue going on. Perhaps your display is not configured correctly?