That every request made through the edge browser for a non local user is associated to his/her live ID is pure coincidence ;)
Sometimes when I look into my dns logs, I cry a little bit inside too.
but firefox has different dns requests going to their servers as well.
If you want to block it from things you don't trust to have such methods (or always listen to them) you'll have to upgrade to a firewall that can filter outbound connections to IPs the client hasn't received a DNS response for or require use of an explicit HTTP proxy for outbound connectivity.
Just blocking DNS can be a good middle ground for reasonable effectiveness without as much effort.