Why do 1080p screens have a pixilated appearance, unless the monitor displays a resolution of 4k when the retina has a resolution of 1080p?

I had a 1080p monitor and it looked slightly pixellated. I've bought LG Ultrafine 4k screen and it looks great. I asked my job to buy a monitor for the office and they offered me a 1080p monitor. I am a web and graphic designer, so it is important for me to look good. I've also noticed that the native display is 2560 x 1600, which is not 4k. Am I going to have to convince them to get a cheaper 4k monitor? I have a Macbook Pro Touchbar 2017.