The majority of the monitors released in at least the last 5-7 years should have the native white point at 6500K, unless it's some specialized display, which usually cost much more. If the native white point of your display differs from D65, then there might be something enabled in the settings, like cinema or gaming mode. Before calibrating and profiling any monitor, always turn off all enhancements first!

Unless your goal is hard proofing a printed copy in an environment with calibrated light source, always calibrate your display to D65. In printing industry your light source and monitor would both be calibrated to D55 (5500K), so you could directly compare printed hard proof to the original displayed on the monitor.

Human eye itself is more or less calibrated to D65, that's why we perceive direct sunlight at noon, which is ~5500K as warm coloured. Our vision system registers the shifts in the colour of the lighting, but our brain psychologically adapts to the changes and we still refer to the light as white unless it's colour temperature drops as low as 3400-4000K. This psychological adaptation is what makes us remember this different degrees of warm white as simply white, same as 6500K.

Particular colour hues, on the other hand, are often remembered as we see them, disregarding the white point shift to some degree. That's why calibrating to D65 is essential for any colour critical work - so you would remember the colours correctly and your vision system won't have to adapt each time to different white point.