Let's say I want to follow another friendly civilization's army stack, to know where they exactly are right now. Is there a way to issue such an order to one of my units, or do I have to follow them manually every turn?
I don't believe there is such a thing. The reason is probably that it's not just a convenience function - having such an order is a tactical advantage.
In plenty of cases a weaker unit can move beyond visual range of a stronger enemy unit, forcing the stronger unit to guess which way it has gone if it wants to catch up and attack it. This is particularly true at sea, or when the weaker unit is in its own territory on a road, or is an explorer.
If a 'follow' command existed then this would change the gameplay, preventing the weaker unit from escaping.