The D610 is a 24MP camera.
The D800 is a 36MP camera.
When you are viewing images from both at 100%, you are magnifying the D800 images by a larger enlargement ratio than you are magnifying the images from the D610.
If you are viewing the images on a 23" HD (1920x1080) monitor, you are enlarging the D800 images to the same scale as a print about 77 x 51 inches. With the D610, your enlargement ratio is equivalent to a 63 x 42 inches. In terms of area, the D800 image is being enlarged by 48% more than the image from the D610! This also means the area of each sensel (pixel site) of the D610 is 1.48X the area of each sensel on the D800 sensor.
The higher enlargement ratio allows you to see the detail of the smaller pixels on the D800 sensor at the same size per pixel as the larger pixels of the D610. With smaller sensels (pixel sites), Poisson distribution noise, caused by the random distribution of photons, will be more noticeable.
If you normalize both images to the same display size you should see much less difference in the noise between the two respective sensors.
The D610, a camera introduced in October 2013, might also be expected to have a little more advanced sensor design and better noise reduction than the D800, introduced twenty months earlier in February of 2012.
Beyond that, the sensors in the D800 and the D610 might not be of similar design technology. Some cameras have sensors that do more on-die noise reduction while other cameras have sensors that do less on-die noise reduction and apply more noise reduction in the off-sensor processing pipeline.
Both the Nikon D800 and the D610 have Nikon Expeed 3 image processors. Both cameras have 'Sony' sensors (although it must be remembered that just because a sensor is manufactured by Sony Semiconductor for Nikon does not necessarily mean a sensor was designed by Sony Imaging for Nikon).
The example images also appear to be moderately underexposed, which will increase the noticeable amount of noise by decreasing the signal to noise (S/N) ratio. When read noise is constant and less light is captured in the exposure, the noise is a higher percentage of the total signal coming off the sensor.
Compare the D800 and the D610 at DxO Mark (the D600 is also included in the linked comparison):
With the size normalized (the 'Print' specification), there is virtually no difference between the two cameras with regards to S/N ratio or dynamic range.
Yet when we change 'Print' to 'Screen', in which the size is not normalized, we see a more noticeable difference:
The exact values for the D610 at ISO 1600 when we hover over the chart:
- DR: 10.45 EV
- S/N ratio: 29.3 dB
The exact values for the D800 at ISO 1600 when we hover over the chart:
- DR: 10.03 EV
- S/N ratio: 27.7 dB
A change of 1 EV is one stop of DR.
A change of 3 dB is one stop of S/N ratio.
At ISO 1600, the D610 is roughly 1/2 stop better in terms of DR and and 1/3 stop better in terms of S/N ratio compared to the D800 when images from both are viewed at 100%. This is pretty close to what would be predicted by a camera having sensels 1.5X the areal size of the other.