Is "user-select: none" bad UX?

I was having a discussion today with a colleague. I showed her "user-select: none" - a CSS property that stops a user from selecting text.

To me, this property allowed a more app-like experience to be created on a page - ensuring that events like double -clicking didn't cause text selection on areas that wouldn't be selectable in an app (like options on a nav bar, custom select elements, labels, etc).

My colleague disagreed that it added any polish, and suggested that users should be able to select and copy any text in a web-app.

What's the consensus on this from a UX perspective. Should we be trying to make sites feel more like apps? Or should the web feel like the web - meaning that we don't change default behaviors like this one?