Move tool is different

When the Move tool is active, look for the Auto Select setting in the options bar at the top of the Photoshop workspace. You probably want to change the Auto Select setting until it does what you want. If Auto Select is on, the Move tool will drag the layer or layer group under the pointer. If Auto Select is off, the Move tool drags the layer selected in the Layers panel.

Just adding to the previous answer... in recent versions of Photoshop Auto Select defaults to on, and that is the behavior you are describing. It's been there for years but always defaulted to off.