Many Macs are not compatible with Mavericks, which is Apple's latest operating system for Macs. Here a support article with the system requirements for Mavericks: http://support.apple.com/kb/ht5842
Now let's get back to the Apple ID issue.
What this issue is mainly telling you is that that Apple ID you used doesn't have Mountain Lion in your purchases. If you actually did use your brother's Apple ID there shouldn't be any problem. You can contact AppleCare (if your Mac is still covered under the warranty, if not there may be a charge) to talk about this and they will give you many solutions. I provided some solutions. To chat with an Apple Express Lane Specialist, go to http://www.expresslane.apple.com (again, it is complimentary if your Mac is currently under warranty).
Apple's number is : (1-800-MY-APPLE)
If you erased your hard drive, you should never erase the Mac OS X Base System since that has your installation information. There is maybe a way to reset that information but when you are booted onto OS X Utilities there is no finder application installed, it's nearly impossible. Like I said, try calling APPLE.