Do merc spires or spires of another faction under my control count as faction spires?

Yes. A Faction Spire is a Spire that is under the Control of your Faction. However, just like with Units, they are still not considered for effects that specifically refer to your Faction. For example, if the Grovetenders construct Babel, a Merc Spire, and also have the Fortress Advancement Spire Precision which says that attacks made by Grovetender Spires deal extra Damage when they roll blanks, this advancement has no effect on Babel. In Narora solo Chapter 1, Saints Will Persevere, the Narora have Control over multiple Grovetender Spires. These Spires are considered Faction Spires, but are not considered Naroran Spires and will not increase the CP for the Naroran player.