Display
The Xperia Z5 comes with a 5.2-inch LCD, possibly IPS in technology, featuring a resolution of 1920 x 1080. Sony has used a 1080p display in the 5.0-5.2" size class since original Xperia Z, and although the quality of their newer displays is obviously superior, Sony is clearly happy with the Z5's pixel density of 423 ppi.
In some ways Sony has done the right thing by resisting the urge to include a 1440p display with the Xperia Z5, instead offering a higher-end phone (the Xperia Z5 Premium) for those that want a larger, 4K display on their device. 1080p is just fine in the vast majority of circumstances, especially when it uses standard LCD subpixel arrangements.
In general, the Xperia Z5 is a sharp, crisp, bright display with decent viewing angles for an LCD. Maximum brightness was particularly impressive at 600 nits, and the unnoticeable piece of glass between the display and outside world makes the display easy to view outside in strong backlighting. Auto brightness was responsive as well, and manual brightness allows you to turn the brightness down very low for reading in dark environments.
Where the Xperia Z5 falls down is in accuracy pretty much across the board. By default, the display is heavily tinted towards the blue end of the spectrum, which not only gives whites that cold hue, but also drags colors away from their true value and towards blue. Black levels are mediocre, leading to a contrast ratio of 1140, and there's noticeable backlight bleed along the top edge of my retail unit in some situations.
Measured with an X-Rite i1Display Pro, using SpectraCal's CALMAN 5 software, the overall color gamut of the display is well outside sRGB. Normally this would be fine, but Android lacks any sort of color management, which gives the Xperia Z5's display an oversaturated appearance (and this is without turning on the ultra-vibrant display mode). Reds are oversaturated the most, which makes them 'pop' at the expense of accuracy, though skin tones are reasonably okay.
Accuracy gets even worse when you're viewing images with the X-reality Engine enabled. An oversaturated display can look good in some situations, but with the X-reality Engine enabled, photos and videos viewed in the stock applications have an unnatural color reproduction that just looks strange. I'd recommend turning off the Engine immediately.
It is possible to correct the accuracy of the display somewhat by adjusting the white balance in Sony's pretty decent control screen. Setting reds to +205 and greens to +140 brings the color temperature from 9800K to 6500K, and addresses the blue tint throughout the display's color reproduction, improving accuracy significantly. Unfortunately, the by-product of making this change is a reduction in peak brightness, so it's by no means perfect.
The Xperia Z5 also comes with a glove mode, which I'd recommend enabling only when you plan on wearing gloves, as it increases the touchscreen's power consumption.