What is the relationship between these two different methods of color correction?
In Photoshop and similar raw processing applications the Color Temperature adjustment is along the Blue←→Amber axis. The Tint adjustment is along the Green←→Magenta axis. When displayed on a color wheel, the Blue←→Amber and Green←→Magenta axis are displayed at 90° to each other.
When cameras have a built-in WB color correction setting, such as the one that is illustrated in the question which is included in Canon cameras, the full extent of the maximum adjustment is much smaller than the full range of the in-camera Color Temperature adjustment as well as the CT and Tint adjustments provided by raw converter applications such as Photoshop.
Some raw processing applications, such as Canon's Digital Photo Professional 4, include the same fine adjustment tool as provided in many cameras (with gradations 1/10 as fine as each in-camera unit). This tool allows for more convenient very fine adjustments along each axis than the CT and Tint adjustments. It also allows the application, when opening a raw file, to automatically apply and display the in-camera settings for WB correction at the time the photo was taken.
With Canon cameras, which is the source of the illustration in the question, each unit of adjustment along the Blue←→Amber axis in white balance correction is equivalent to a five mired color correction filter. The fullest value allowed, nine units, would thus be equal to 45 mireds. This is roughly equal to about 1/3 the correction provided by a full CTO (+137 mireds/orange) or CTB (-137 mireds/blue) color correction filter. 45 mireds is a fairly small distance along the entire Blue←→Amber Color Temperature axis. The difference between 2,000K and 10,000K is 400 mireds.
Technically speaking, the mired scale is strictly along the Blue←→Amber color temperature axis because the units are defined using reciprocals of values expressed in degrees Kelvin.¹ Although Canon does not specify the value of each unit along the Green←→Magenta axis in the in-camera White Balance Correction setting, one would assume it is based on the same amount of perceptual difference along that axis as the 5 mireds of adjustment per unit provides along the Blue←→Amber axis.
What, you ask, is a mired? From Wikipedia's Mired entry:
Contracted from the term micro reciprocal degree, the mired is a unit of measurement used to express color temperature. It is given by the formula:
where M is the mired value desired, and T is the color temperature in Kelvins.
The use of the term mired dates back to Irwin G. Priest's observation in 1932 that the just noticeable difference between two illuminants is based on the difference of the reciprocals of their temperatures, rather than the difference in the temperatures themselves.
As the formula above indicates, a mired is the reciprocal of one megakelvin. As Priest noted, the advantage of using this scale is that the steps, unlike the kelvin scale, are perceptually equal to human vision. A 100 mired adjustment at any point on the scale will appear to shift the color by the same amount (this is not the case with the Kelvin scale).
One specific application of the mired scale is what we commonly refer to as a CTB (Color Temperature Blue) filter, which is the needed correction to filter a 3200K tungsten light to match ambient daylight of 5700K. In such a case the filter would be placed over the tungsten light and not the camera's lens.
The way corrections are calculated is based on two logarithmically spaced Kelvin scales running in opposite directions with a single mired scale between the two.
As illustrated below, a Wratten 80A filter, which has a value of -131 mireds², can convert a 3200K tungsten light source to about 5500K. Notice that the difference on the Kelvin scale between 3100K and 3300K is the same as the distance between 5200K and 5800K on the opposing scale.
In the neighborhood of 3200K, a 5 mired blue filter would equate to a shift of about 300K. But at 6000K, a 5 mired blue filter would equate to a shift of only about 100K.
¹ At the time the Kelvin scale was developed as a way to measure the "hue" of light emitted from light sources, there were no artificial light sources in existence that strayed very far from the light emitted by black body sources of radiation at various temperatures. This is no longer the case. Many modern light sources (fluorescent lighting, LEDs, high pressure sodium vapor, etc.) emit light at hues that are a large distance, along the Green←→Magenta axis, from the Blue←→Amber axis that is defined by the hues of light emitted by black body radiators at specific temperatures measured in degrees Kelvin.
² That is, -131 mireds is 131 mireds of blue correction. A filter with 131 mireds of amber correction would have a value of +131 mireds.