It's roughly true that light sources with the same color temperature have the same appearance. In fact, matching light sources in this way is exactly the reason we use the Kelvin WB scale in photography. However, there are three big caveats.

First, there's also a magenta-green axis

Human perception of color is complicated. White balance as measured in Kelvin is simplification of one aspect of that perception, basically relating to orange/blue balance. This is reasonably helpful for light sources that approximate blackbody radiation, but doesn't fit for a lot of artificial light sources, which may tilt more towards green or magenta — tints which are off the Kelvin WB scale. See How does Kelvin for color temperature relate to Kelvin for actual temperature? for details on the Kelvin scale, and Relationship between tint-temp and magenta-green-blue-amber white balance corrections?

Second, not every light source covers a complete spectrum

Sunlight filtered through the atmosphere, or the candlelight you mention, or an incandescent bulb — all of these have a clear weight on that Kelvin scale, but they also put out light across the visible spectrum (and into the invisible infrared and ultraviolet). This is not the case with gas-discharge or fluorescent light sources. That includes sodium-vapor streetlights, fluorescent bulbs, and LED lighting.

Our perception of color depends on wavelengths reflected by the lit objects in the scene. If those wavelengths are reduced (or entirely missing) in the source light, they can't be reflected, and that alters our perception of color.

For more on this, see What is Colour Rendering Index (CRI)? — and Can I color-balance a photo if I know the light source within the photo?, where someone wants to balance color but is limited by the narrow-band light source.

Third, the numbers are nominal.

No candle flame snaps to exactly 1800K, and the color of sunrise and sunset is so complex that it's probably safe to say that literally every one is different. The labeled values on lights are not precise — probably more so for gear designed for photography, but consumer light bulbs will vary quite a bit from what the box says (as well as from brand to brand).

Sooooo.....

You ask:

Could you say that the values of the pixels are the same with these three sources of light?

And in practice, no, this is completely unlikely.

They may, however, be similar enough that they work together in a single photograph without causing the disruptive look we get when one area of the photograph is cool blue and another quite orange due to mixed lighting.

In your example of a bulb rated 2800K and a sunrise or sunset coming through a window (nominally 2400K), the window light may look a little warm (that is, warm in the artistic rather than physical sense: more orange) in your photograph balanced for the 2800K bulb — but then, that may be exactly what you want.