"Upscaling" means enlarging the image to fit the display.
"Interpolating" means some sophisticated algorithm is in play to smooth out jaggedness you get from non-integer upscales. e.g. 1440p to 2160p. If you do the simplest upscaling without interpolation (nearest neighbor) with a non-integer scaling ratio, it looks pretty bad as some pixels are doubled, while others are not. Interpolating means each source image pixel influences the same number of screen pixels regardless of scaling ratio.
https://en.wikipedia.org/wiki/Comparison_gallery_of_image_scaling_algorithm
Unfortunately, a side-effect of interpolation is that the image looks softer (it's really not, it's just that it looks soft compared to the razor-sharp pixel edges on a LCD display - more on this later). For this reason, often people prefer nearest-neighbor scaling when upscaling 1080p to 2160p (4k). Each source image pixel gets mapped to exactly 4 screen pixels.
Most monitor manufacturers know this and will revert to nearest-neighbor scaling when integer-scaling. Some will even give you a choice of type of scaling algorithm to use. In the old days, bilinear was the most common type of scaling interpolation because of its computational simplicity. Nowadays, the processors built into monitors are easily able to handle bicubic interpolation. A few have even moved up to lanczos interpolation. Those are the words you should look for in your monitor settings to indicate a scaling option.
As for razor-sharp pixel edges, they're actually high-frequency noise which distorts the image. Pixels aren't supposed to be square. They're supposed to represent the color of an infinitesimally small point in the actual image. When enlarged they're supposed to be a blob whose brightness falls off the further you get from the center (a gaussian point spread function). Pixels on the old CRT monitors were like this, so you could run them at any resolution and they'd look equally good. But LCD monitors have square pixels, which means only resolutions which perfectly align with the pixel edges look good. The sharp edges and corners add noise to the image - information which doesn't really exist. It's just that if the image being displayed has horizontal or vertical lines (like computer UI windows and menus), that noise happens to coincide with the details in the image being displayed. making the overall result look much sharper that should be possible at that resolution.
A consequence of this is that if you're viewing the YouTube UI and video at 1080p, then view the two at 4k (with a 1080p video upscaled), then the video will look worse because it doesn't benefit from the artificial sharpness the horizontal and vertical lines in the UI elements enjoy. It's possible to augment sharpness in video through something called unsharp masking, which is an optical illusion which tricks your brain into thinking an image is sharper. But computer monitors and games generally eschew it. The sharpness setting on your TV does it though - degrades the image by increasing unsharp masking, tricking your brain into thinking the image is "better".
https://en.wikipedia.org/wiki/Unsharp_masking
So try to find an image scaling option in your monitor settings if you're displaying full-screen 1080p content (generally games run in full-screen mode - there's about a second delay in a black screen as the monitor switches resolution). YouTube videos are generally scaled by the browser. I'm not sure if the browser does it itself, or if the site instructs the browser on what scaling algorithm to use. So you might want to try viewing YouTube in a different browser and see if it looks better/worse. Likewise, video in a window like a video player will be scaled by the player using whatever algorithm the video player chooses. So try a different player (or see if the player has different scaling options).
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