It should actually look a bit better than native 1080p, especially if we are talking about video from streaming services, which tends to be heavily compressed to save bandwidth. Youtube, for example, puts relatively heavy compression on their 1080p video, making it appear somewhat blurry when played full-screen on a 1080p display.
For 4k vs 1080p streamed content, I agree.
But for full res video with minimal compression, the difference is mostly illusory. When a studio downsamples a 4k video to 1080p for distribution, they do it using a high-quality encoder which takes on the order of 2-10 hours per hour of video to downscale, figuring out the best way to mash 4 pixels of data down into 1.
When you play a 4k video on a 1080p screen, the image is downscaled in software, or a scaler built into the GPU. These are designed to work as quickly as possible since they have to work in real-time. It's done by a
generic algorithm which doesn't at all consider what exactly the image is trying to show.
The reason it looks sharper is an optical illusion. Your eyes have special cells which sense edges (borders of light vs dark areas). They trigger when they see a border, and your brain gets excited that it's seeing an edge. That's how
unsharp masking works - it actually destroys information in the image, but exaggerating the borders between light and dark make your brain think the image is sharper and thus better looking. (The sharpness setting on your TV does unsharp masking.)
Downsampling a 4k video to 1080p with a simple scaler results in a slight false sharpening effect. The image quality is actually slightly worse than 4k video downsampled to 1080p properly with a good encoder. But because of the false sharpening, the edge detectors in your eyes get your brain all excited into thinking it's seeing a better image. If you applied a slight unsharp mask to the studio-produced 1080p video, it would appear slightly better than the 4k video downscaled to 1080p in real-time. (This is why your TV has a sharpness setting. Video is transmitted in a format intended to convey the most information, but that ends up looking slightly blurry. The TV adds a bit of unsharp masking to make the picture "look good" even though it's destroying a bit of info in the process.)
Of course, if your video player doesn't allow you to add a bit of unsharp masking, you may in fact prefer the downscaled 4k video over a properly encoded 1080p version of the video.