How does Netflix prevent users from taking screenshots of chrome browser?
Most streaming media services now make use of EME https://en.wikipedia.org/wiki/Encrypted_Media_Extensions. The media players built by these services make use of EME to invoke the underlying DRM (Digital Rights Management)
WebBrowser -> HTML5/Javascript -> EME -> DRM
And yes, of course you can build your own solution using EME.
To add to the other answers and comments - the screen capture prevention mechanisms are actually dependent on the DRM security level and the device capabilities and so may be different on different machines.
Browsers using a SW based DRM solution which is not linked in to the secure media path on the device actually will allow screen capture.
Browsers using a HW based DRM or a SW one which is linked to the device secure media path will prevent screen capture.
Typically streaming services restrict their high resolution content, e.g. 4K, and sometimes even their high value content, e.g. live sports, to devices which support a secure media path. You can see this with popular streaming services where you may be able to stream a video in 4K on one browser/device combination but only in 720p on another, even on the same device.
Content security is an ever changing domain so you may find a particular browser and device combination supports different security levels over time.