The moon is already a little reddish
The moon does significantly redden when it's close to the horizon, especially if you can see it over the ocean or from very flat ground, where you have a clear view all the way to the horizon. But even at best, it's more an orangening. Why is the moon's reddening less obvious than the sun's?
I think it's important to think about what 'reddening' means. You said "the spectrum of moonlight is more redshifted than that of the sun, which should contribute to an even more intense reddening" -- but that's actually the opposite of the case. (I would not use the term "red shift" in this context; that term has a specific astronomical meaning related to relativistic speeds.)
The sun at the horizon turns red because the blue light is being refracted around the atmosphere (contributing to blue skies for the day side of the world), and the red passes straight through. Consider if the sun produced only red light -- then you would see no additional reddening near the horizon because there's no blue light to remove from it.
So if the moon's spectrum is already redder than the sun's, then we would expect its reddening to be less intense than the sun's, not more -- there's not as much blue light to remove, so the change is smaller.
How red is the sun, really?
To be fair, not every sunset features a really red sun. A middling orange is far more common than the deep crimson that you usually only see when there's a lot of dust in the atmosphere, such as from a recent volcanic eruption.
But also, photos are tricky things, because photographers almost always adjust the color of a picture, and the display unit you're viewing it on will alter the perceived colors. Moon photos don't typically try to enhance the moon's redness -- for example, the photo you posted is clearly meant to retain some realistic green color in the trees, which can't be compared to a sunset photo that's been tweaked for maximum color depth.
Your eyes aren't perfect either.
It's possible that another contributing factor (though I'm just speculating) is that the moon is much less bright than the sun, and seen against a generally dark sky, and under those conditions your eyes do a worse job of color discrimination. Your eyes' color-vision cone cells are less able to function with lower light levels and extreme contrast, so your rod cells take more of the load and you see colors as more washed out.