For example, the color temperature of a candle flame, sunrise and sunset have the same color temperature, knowing that its color temperature is the same, could you say that these sources are similar or equivalent? Could you say that they would produce the same picture of a scene? Could you say that the values of the pixels are the same with these three sources of light? Thanks in advance

It's roughly true that light sources with the same color temperature have the same appearance. 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.

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.

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.

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.

It wouldn't be accurate to say the sources are the same but it would be accurate to say the they are similar. Atmospheric differences (moisture, dust, etc.) can change the color temperature of the sun not just from day to day, but even from minute to minute.

Since lighting conditions are often dim, if you want to photograph a subject in these conditions, a flash might be used. You can use a CTO gel on the flash (CTO = Color Temperature Orange) to bring the color of the flash closer to the color of the candlelight or sunlight so that any color-adjustments performed in post processing wont have radically different color temperatures. But even the CTO gels come in 1/4, 1/2, 3/4 and full CTO strength (depending on whether you need a pale yellow/gold vs. an intense orange).