Hi @Emirhan depending on how your sound is set it should work. Xbox has specific setup instructions and if you're using a soundbar or some other external sound source it's not going to work. I use a soundbar and "muting" shows up on the screen when I ask the Nest device to mute, but it does nothing to the volume.
Volume control used to be easier with the Kinect device and Cortana, but that has been discontinued.
Are you in any of the betas for Xbox? I'm in Alpha-Skip ahead and it keeps busting things with every update. Then it's fixed and then poof busted again.
Even if not in one of their betas, their normal software updates can break stuff. MS may also have a glitch in their API or may not have kept up with changes Google has made. Microsoft will have to support why it's not working -- assistant failure is the symptom, not the cause. There's also a specific box in settings on the console that needs to be ticked as well, so that could be botched.