This answer applies to all AoE 2 variants, from the original to the Conquerors expansion to the HD release. There is no attack-move in AoE 2. Right-clicking on an enemy unit will order your military units to pursue and attack that unit, ignoring everything else along the way. If the target moves out of line of sight or dies, your military units simply stop where they are and return their stance behavior. For this reason, it is inadvisable to right click on any target not in range of your units, as your units will spend most of their time walking and accomplish very little before dying.
AoE 2 does have a mechanic that is similar to attack-move, called patrol. I believe you have to turn on advanced commands (F4 by default) in order to enable it. I like to call patrol "attack-move-in-circles", because that is essentially how it works. The patrol command lets you set waypoints with shift click, and is finalized with a right click. Once you do that, your units will "attack-move" along that route, returning to their starting location after the final waypoint.