Could I have a virus or malware on my Xbox

No. You can't get a virus on the console.

Both the Xbox One OS and Edge on Xbox One are in lock down. You could go to the most nastiest virus, malware infested website on the web and your console still won't get a virus.

Yes, these are server side issues, bugs as others have states, not malware or a virus.

A computer virus, key logger, malware perhaps?

The Xbox one can not get malware or viruses on it, but the web browser could have browsed to a website that was "infected" and was designed to look like a virus popping up on the screen - or even just an ad or cookie or some kind of content from a regular
website that can technically be recorded as the xbox visiting it since it got data from it.