If we assume that you can use it as a mount at all, the list of restrictions for the hunting dog is extensive and excludes many of the following mounts:
It can move through other creatures and objects as if they were difficult terrain. The hunting dog suffers 5 Power Damage when finishing his turn in an object.
At the beginning of his turn, the dog automatically knows the position of his target. If the target was hidden, it is no longer hidden from the dog.
The hound will appear in an unoccupied area of your choice within 30 feet of the target. Role initiative for the dog. In his turn, he can only move straight to his target and use his action to attack his target. The hound can perform occasional attacks, but only against its target. In addition, while the dog is within 5 feet of the target, the goal is to keep throws against a spell you cast. The dog disappears when reduced to 0 hit points when its target is reduced to 0 hit points or after 5 minutes.
- it can only move on the most direct route to the destination
- It can only use its action to attack its target, and it can only perform occasional attacks against its target
- It only takes a maximum of 5 minutes
To answer your question, no, it does not move as you guide it, but straightforwardly through objects and creatures, if necessary, towards the target and can only use its action to attack the target.
A more liberal DM can also allow him to stalk his target if it's too far away to attack it … but that's because of the DM.
You may not be able to use it as a controlled mount anyway
Controlled mounts have this requirement:
You can only steer a mount if it has been trained to take a rider.
Your dog is not trained to take a rider, so you can not control him.
It is likely that your dog is intangible and unwilling to be a mount, even if it is tangible, and therefore can not be mounted effectively
It must be for a creature to be a mount at all
A willing creature that is at least one size larger than you and has an appropriate anatomy can serve as a mount according to the following rules.
When you summon the dog
summon a howling creature of darkness to harass your enemies
Move through other creatures and objects as if they were difficult terrain
Which strongly suggests that it is intangible and therefore can not be mounted. So, while it is one size larger than you (small in size, medium in size), an immaterial creature does not have the anatomy relevant to its assembly.
Moreover, the dog's description is not well suited to being interpreted as a "willing creature" that can be mounted, even if it is tangible.