How does HDR relate to color space and bit depth?
TV technology and video encoding is in principle outside the scope of this site. 
Still, as photographers we are presumably interested in new display standards that promise both more dynamic range and wider color gamut. HDR computer monitors are coming, so it probably won't be long before photographers too will be interested in HDR displays.

HDR by itself just means High Dynamic Range. Dynamic range is the brightness ratio between the brightest and the darkest parts. "High" is relative to what the sensor can capture or to what a standard display can show.

HDR video is one of several standards for video encoding. They have names like HDR10, Dolby Vision, and Hybrid Log-Gamma. They all support higher bit depth (10-12 bits), wider color gamut (Rec. 2020 and/or DCI-P3), higher dynamic range, and 4k resolution. They are not compatible with each other, but some devices support more than one HDR standard.

"HDR TV" doesn't necessarily mean much, except that it will accept a HDR video signal and display an image. It does not necessarily mean that it can display the wider gamut and higher dynamic range, just that it can read the signals and show something, even if what it shows is plain sRGB with a mundane dynamic range.

"UHD Premium" is a certification standard for displays that can actually show HDR. Requirements include:


minimum 10 bit color
minimum 90% of the DCI-P3 color gamut
use a HDR transfer function (The transfer function for sRGB is called 'gamma', it translates bit value to brightness value. HDR uses a different function so it can cover a larger dynamic range with a small increase in the number of bits)
either: Minimum 1,000 nits brightness and max 0.05 nits black level, contrast ratio 20,000:1 or better (for LCD displays)
or: Minimum 540 nits brightness and max 0.0005 nits black level, contrast ratio 1,080,000:1 or better (for OLED displays, they can't go as bright but have better black levels)


Standard displays are 200-400 nits and rarely go beyond 1,000:1 contrast ratio. So a "Premium" HDR display has significantly higher peak brightness as well as an order of magnitude better contrast. 

Brighter highlights, higher contrast ratios, and a different "bits to brightness" encoding/decoding all combine to enable the "high dynamic range" part of a HDR TV.

Color space and bit depth is only related to HDR in the sense that they are part of the same standards. Yes, you need more bits to represent a higher dynamic range without banding. But the larger color space is a different decision, it's more a matter of including enough improvements to make a new video standard worth the trouble. (4k is also part of the standard, and 4k is not related to HDR either.)

"The sun looks brighter on a HDR display" means that a) the display emits more light in the brightest parts of the scene, and b) it may look even brighter because the rest of the scene can be dimmer without losing detail. 

So it's a real improvement; the wider color gamut and higher dynamic range can  contribute to more vivid photos and video. 

The caveat is that TVs/monitors may be sold as "HDR compatible" without any actual HDR capability in the display. The content needs to be made specifically for HDR to make use of the new capabilities. And for computer monitors, you need a graphics card that supports HDR output.