Yes if the mount is an intelligent ally. Otherwise no.
There isn't a hard and fast rule for this, since the rules for building combat encounters are just guidelines to begin with. With that said, I think the important question here is whether the mount is simply a tool that the player uses to enhance their movement, or an intelligent ally who will move, attack, and make decisions for itself.
For the specific example of siccing a mastiff mount on an enemy, if you allow that to happen then the mastiff is an intelligent ally and should count. However it's worth noting that by the rules, this isn't something you can normally do with a mount. Using the normal rules for mounted combat, a non-intelligent mount can only move and take the Dash, Disengage, and Dodge actions.
Even with animal companions, who are notably more intelligent and obedient than a regular pet, they generally aren't considered able to take action on their own. With that in mind, it seems that DnD generally assumes that only very intelligent (usually magical) creatures can act completely independently.