Armor Stands will not truly face their target the first time a command to do so is given to them. If they were told to look at an entity's feet/eyes the hit-box will face towards them horizontally, but the actual model will not, nor will they face them vertically.
To make the Armor Stand face an entity proper you have to first, tell it to face an entity's feet or eyes, and then tell it to face the opposite of what was previously chosen. It will not matter if it is a different entity or not, so long as it is facing the opposite area of the body that it was previously told to face.
However, once properly facing what is commanded of it the second time around, it does not look at the right place. Looking at an entity's eyes causes it to look a block above their head and looking at an entity's feet causes it to look them in the eyes.
Also, the solution of commanding the Armor Stand to face the opposite area does not work at all in chain or repeat command blocks. As it will always have it's hit-box face the target horizontally, but not vertically, nor will it's model change to face the target.

Blue Command: summon minecraft:armor_stand ~ ~0.5 ~ {NoGravity:1b,CustomName:"

",CustomNameVisible:1b,ArmorItems:[{},{},{},

]}

Yellow Command: execute as @e[name=Laser] at @s run tp @s ~ ~ ~ facing entity @e[type=villager,limit=1,sort=nearest] feet

Green Command: execute as @e[name=Laser] at @s run tp @s ~ ~ ~ facing entity @e[type=villager,limit=1,sort=nearest] eyes

Yellow Command: Laser is told to face Villager's feet
Model doesn't move, hit-box does, but only horizontally

Green Command: Laser is told to face Villager's eyes
Model moves and hit-box moves both horizontally and vertically.

Yellow Command: Laser told to face a different Villager's feet
This time the model and the hit-box have moved as they should have done the first time. Though, they aren't facing the feet of the Villager

Attempting to chain the Blue, Yellow, and Green Commands doesn't align the armor stand properly and only causes it's hit-box to face the horizontal direction of the target.

I'd also like to ask that this related bug be looked at: https://bugs.mojang.com/browse/MC-133332