Can I use The Syren to target my teammate's unit?

No, you cannot use The Syren to move your teammate's unit.

According to Rule 484.8.c.2 and Rule 484.8.c, while "friendly" in 2v2 (Magma Chamber) generally includes both you and your teammate, control is not shared. Specifically, you may not issue movement to your teammate's units.

Furthermore, a RiftJudge FAQ (ID 5119) clarifies this interaction explicitly:

• Ruling: You cannot move an ally's unit to your base. The effect of The Syren moves the unit to its controller's base (the base of whoever currently controls that unit).

In a 2v2 game, if you activate The Syren targeting a friendly unit that you control, it will move to your base. You cannot use it to move a unit controlled by your teammate.