How do I connect user account with Time Machine back up?
I saw your question before, but was at a loss to provide specific information. However, I took a look at Pondini's OSX and Time Machine Tips.

In particular on the page, How Local Backups are Stored, I noted that it's actually the hardware address, not the user ID, that is used as identification by Time Machine. From the page:


  Each Mac is identified by a hardware address internal to the Mac.   It's embedded in the Logic Board (motherboard).  The hardware address isn't displayed in the backups, but is kept in an extended attribute of the folder representing the Computer.
  
  That's why when you get a new logic board, Time Machine thinks you got a new Mac.  That's also why you can rename your Mac (System Preferences > Sharing) and Time Machine will automatically rename the Computer folder on the backups;  or you can have two Macs with the same name, but Time Machine still knows which backups belong to which Mac.   


However, on the same page, Pondini goes on to say that each backup volume is assigned a Universally Unique IDentifier (UUID) that is in the extended attributes of the volume. Pondini says that this "should" cause Time Machine to automatically recognize the old backup.

However, if Time Machine does not automatically recognize the old backup via the UUID, then there is a procedure for manually restoring the association.

It's getting fairly involved at that point, and though it might be worth the effort from an investigative standpoint, you might be better off starting your Time Machine backups from scratch (just to wipe the slate clean, so to speak).