Use Magic Trackpad 2 with Google Chrome

No, it can only be paired to one computer at at time according to Magic Trackpad Manual. You would need to un pair and then re pair it to another computer in order to use it on both computers.

Here's the extract from the manual:

After you pair your Apple Magic Trackpad with a Mac, you can pair it again with a different Mac. To do this, you first remove the existing pairing and then pair the trackpad again.

Your trackpad preference page is an earlier version than 10.8. I think 10.7 or before, so this is likely a software issue** The 10.8 page has changed a lot compared to your screenshot.

In the lower left it should say trackpad battery level and have battery icon with percentage. Having had some bluetooth problems in the past it should give you a battery level even if it is zero, unless the device is not completely paired, possible hardware issue.

With 10.8 under the Bluetooth preference page it will give you a list of all Bluetooth devices that have been paired with your machine, at the bottom below the devices should be a + - and a sprocket with a drop down menu that includes connect, rename, add to favorites, update name, update services, and show more info. The - will delete the selected device, I would suggest you delete the ghost mouse and maybe even the trackpad and try to pair the trackpad again. Also on the bluetooth page I have discoverable not selected.

If you are running 10.8 and you have Apple Care I would contact Apple as your screen shot is showing an earlier version than 10.8 as far as I can tell.