Or this scene from the Battleship movie?
https://www.youtube.com/watch?v=5AJKah6-YCYYeah, it doesn’t work like that. No anchor could handle the momentum of a ship underway like that. Standard practice is to come to all stop prior to deploying an anchor. After deploying you would back astern to set the anchor.
Back in my active duty days I learned that it wasn’t so much the flukes of the anchor digging into the sea bed that held a ship in place as it was the weight of the chain deployed. Not all anchors have flukes. Submarines have mushroom anchors for instance.
Losing an anchor or dragging is not unheard of. When anchored we were required to regularly take fixes on our position to ensure we weren’t slipping.
Surface ship anchors are visible so if they had a special anchor to cut cables, it would be apparent.