Does playing a unit count as choosing it?

No, playing a unit does not count as choosing or targeting it.

According to Rule 352.4.b, making choices for a unit as it is played (such as choosing a location) is distinct from targeting or choosing a game object for an effect. Furthermore, Rule 352.6 defines targeting as when a card "Chooses one or more specific Game Objects to affect."

When you play a unit, you are placing it into a location (Rule 352.2), which is a game action required to put the unit onto the board, not an effect that targets or chooses the unit.