You are correct. The moon is synchronized with Earth, although not perfectly. There is a wobble which I’ll show you below.
The “face on the moon” means that some people see what looks like a face on the side of the moon we can see. It’s not on the far side, but the side we see.
Here is a picture of the moon with the “face” outlined that people call the “face on the moon.” It is not meant to be comical (though it looks funny), but outlines the moon features people imagine look like a face.
More below the picture.
Now, here’s a comparison of the near and far side of the moon taken by a satellite we placed in orbit around the moon. On the left is the familiar side of the moon we see facing Earth. On the right is the side we never see from Earth because it faces away from us. This shows the moon is tidally locked with Earth so that one side always faces us. More below.
And finally, here’s an animation that shows that the moon isn’t perfectly synchronized with Earth, as you put it (scientifically it’s called “tidally locked”), but wobbles quite a bit from full moon to full moon. Due to this wobble, sometimes you can get a peak of what’s on the far side. You’ll also notice the moon gets larger and smaller in the animation. This variation happens because the moons orbit is slightly elliptical around us, so sometimes it’s a little closer and looks bigger, and sometimes a little further and looks smaller. A full moon when the moon is closest is called a supermoon.