You can't easily tell from software that you have a new board unless the technician failed to properly deserialize the part that was "allegedly" put in. However, the Time Machine change makes me conclude you have a new board.
You could try dumping the hardware identity from the Time Machine backup directory:
xattr -p com.apple.backupd.BackupMachineAddress /Volumes/yourBackupDrive/Backups.backupdb/yourMacName
Since you paid for the repair, Apple should walk you though re-joining the Time Machine over the phone or via internet chat for free post repair.
Did you have it serviced via mail in or at an authorized service provider or at a genius bar? The steps to get repair details can change a little - but you could start by calling AppleCare and asking them to read the technician notes that were left by the repair tech. There should be time stamps when it started repair - possibly a list of parts used, probably some test results they ran, etc... Usually a logic board has it's own internal serialization, but you need to open the hardware to see that or have knowledge of revisions on the boards. If the board was improved after it was made or someone checked the serial number on the board that was fitted in the factory initially, you could know that the one in there wasn't original.
Unless you took a photo of the board that was installed or marked it before the repair, it's unlikely you can differentiate things from software.
Basically, the evidence you can gather is mostly circumstantial unless you are in the forensics trade or work on machines as a trained repair technician that works on that model routinely.