Is this a shutter button on my battery grip?

I think this button is used to take vertical photos. I use this battery grip on a Nikon D5000, I've tried the button it but it doesn't work.

Yes, it should be.

I have a D5000 and I use a similar battery grip. Yes, that indeed is a shutter button!

D5000 doesn't have an internal port for triggering the shutter from battery grip.So, one needs to connect a cable externally to use the vertical shutter. The cable has to be inserted in a port present on the grip and the other end to be plugged into the GPS port of the camera.(I'm sure it must have been provided in the box. I had mine along with the grip and the vertical shutter works great with focus lock and shoot option)

I got a photo of the setup for you. Hope this helps:

The cable is attached on the left hand side of the camera in the photo.

(Please vote if this answer helped you!)