I had this issue my 1st few times replacing customer's screen.
Method #1: Each time I took the home button from the screen they originally had on the iPhone & swapped it with the new home button. That then fixed the issue each time.
Method #2: For the times there was only 1 home button present, I suggested that they 1st backup their iPhone using a computer. 2ndly do I clean wipe of the iPhone. 3rdly restore the iPhone from backup. Which will replace all of your files & settings back into your phone as they were.
Both of those methods worked every time I ran across that issue. However, if neither of the 2 fixes work for you, then YES. It is indeed a hardware part that has been damaged during the screen replacement & you should order a new piece.
If you get the "touch ID" error when installing the new piece it is only due to the old home button being registered to the iPhone's system itself & will be fixed with method #2.
WHEN RESTORING, UPDATING OR FRESHLY INSTALLING IOS ON THE IPHONE, THE PARTS REGISTER TO THE IOS SYSTEM OF THE PHONE. Hope this helps you guys & gals.