The equivalent Canon camera to the Nikon D800 would be the Canon EOS 5D Mark III or 5D Mark IV. The equivalent Canon lens to the AF-S Nikkor 24-120mm f/4G ED VR would be the EF 24-105mm f/4L IS USM.

The equivalent Nikon camera to the Canon EOS Rebel XTi/400D was the D40 or D40X. The equivalent Nikon lens to the Canon EF-S 18-55mm f/3.5-5.6 II was the AF-S 18-55mm f/3.5-5.6G ED II.

Your test is comparing an $800 entry level crop sensor camera from 2006¹ using a $100 "kit" lens to a newer, $3,000 full frame "pro" camera using an $1,100 lens. Which do you think will perform better?

It is possible to get a lot closer to the Nikon result with even the older, lower end Canon camera by using a more comparable lens that allows you to shoot from a bit further back.

• It appears that you were closer to the subject at 55mm (?) with the Canon 18-55mm lens and a little further away at 120mm (?) with the Nikon 24-120mm lens. The different subject distances (how far it is between the camera and the subject) will affect the perspective of the photo.
• The Nikon lens also allowed you to use an aperture as wide as f/4, while the "slower" Canon lens at 55mm restricted you to f/5.6. Assuming you were using the same ISO setting on both cameras, this means you had to use a slower shutter speed (longer exposure time) to get the same exposure value. When shooting with the camera handheld, longer exposure times can increase the amount of blur from camera movement during the exposure.
• The VR (Vibration Reduction) feature of the Nikon lens may have reduced the influence of camera movement while the kit lens that came with the Canon Rebel XTi/400D does not offer IS (Image Stabilization - Canon's VR equivalent).
• You might also need to adjust the in-camera contrast and color settings for the Canon to more closely match what you got from the Nikon, but I think one of the greatest differences may be due to a bit of off-axis light striking the front of the Canon lens and reducing contrast. I'm guessing the Nikon 24-120mm lens had the lens hood supplied with it attached, while the Canon 18-55mm lens that was not supplied with a lens hood did not. One of the primary purposes of a lens hood is to reduce the influence of light sources from just outside of the frame.
• The $1,100 lens on the Nikon is sharper than the $100 lens on the Canon. Canon's EF 24-105mm f/4L IS USM is also much sharper than the kit lens supplied with the EOS Rebel XTi/400D.

I think I had pretty much the same distance in both pictures. I tried to get the same frame, so the focal length is different because of the different sensor size. EXIF says 53mm for the Canon and 105mm for the Nikon, not sure how accurate that is.

53mm on the 1.6X Canon camera is the same angle of view as 85mm on the FF D800. A more proper comparison would have been to use the 24-120mm lens on the Nikon at 85mm instead of 105mm. Then you also need to consider that from the same distance at 53mm with the crop body you need to use an aperture of f/2.5 (something your Canon lens doesn't come close to offering) to get the same DoF you would get at f/4 using the FF camera at 85mm. Since your 18-55mm Canon lens is limited to f/5.6 at 55mm, using f/9 at 85mm with the FF Nikon would give approximately the same DoF as you got with the Canon camera and lens.

In the end, the gear does affect the result here (particularly the sharper, faster lens with stabilization versus the softer, slower lens without stabilization), but a more experienced and skillful photographer can get more out of the older, lower end camera than what we see above. For more along this line of thought, please see: the best way to improve image sharpness on Canon 700D

¹ The equivalent model in Canon's lineup today (which is a much better camera due to improvements in technology between 2006 and 2019) would be the $700 EOS Rebel T7i/800D.