Restoring a MacBook from backup
All of the above are possible.  Thought it is possible to restore from a Time Machine Backup on a Brand new Mac or Newly Installed Drive.


  If I erase the hard drive in Disk Utilities will it think the Time
  Machine back up is a different computer?


This won't matter at all because when you do a clean install and then boot, it will ask you if you want to restore from a Time Machine Backup, another computer, etc.  SO that is completely possible.  

If you just want files, not preferences or user account information or applications, and you can access the drive on the Time Machine and find the backup folder it uses to do the Time Machine Backup, you can just copy over those files. You can very simply traverse through the folders, BUT DON'T MOVE OR DELETE ANYTHING IN THOSE FOLDERS OR ELSE! 

Sorry to use all caps its just important that you don't. I have done this many times. Back up with Time Machine.  New OS comes out.  Erase and Install.  Reset up like a new machine.  Copy over my iTunes and iPhoto and Documents.  JUST DON'T DELETE OR MOVE ANYTHING ON THE TIME MACHINE DRIVE.

If you go the other route and do a restore from a Time Machine Backup, it will just make your computer exactly how it was since it last backed up.  I personally like clean installs but some people don't like setting up a computer from scratch every time and reinstalling things.

Which ever route you prefer.  I hope my answer helped.