Macbook Pro doesn't boot after format and reinstallation

If you erase the drive, then there are no accessible files on the drive and the drive will not boot until you either reinstall macOS or restore from a backup.

Since restoring from a backup gives you the same problems, you will need to perform a clean install of macOS by first erasing the drive again before installing macOS. Then you have two options:

1. You can try to migrate just your user account(s) to the clean install. Do NOT migrate your apps or system settings. This will be the simplest way to transfer all of your data back and minimize the chances of re-encountering the problem. However, this still could transfer the problem back. If you still have the problem, then you will need to use method #2:
2. After the clean install of macOS create a new user account using Setup Assistant. Then connect your backup drive and manually transfer just your personal data to your new user account. This method ensures you are not transferring the problem back again.

With both of these methods you will need to download and reinstall any third party apps and perhaps reconfigure some system preferences.

How to erase the whole physical drive (section#1 with seven steps):

https://support.apple.com/en-us/HT208496

First of all if your backup has the same problem, then even erasing the drive and restoring a defective backup will put you back exactly where you were before. Either you need to restore from an earlier backup which doesn't have the issue, or you need to perform a clean install of macOS by first erasing the drive and then manually copying your data from the backup drive. This will require you to reinstall & reconfigure all of the apps. Maybe you can get by and restore just your user account from the backup while manually reinstalling all your third party apps.

The other thing that may be happening is you may have a defective hard drive SATA cable which is extremely common on the 13" MBPro (mid-2012) model. This cable is inexpensive and easily replaced. You can purchase a cable from OWC.

Can you boot into Safe Mode? If so, then I recommend running DriveDx to check the health of the hard drive. Apple's diagnostics won't catch most drive failures. Post the complete DriveDx report here using the "Additional Text" icon which looks like a piece of paper.

If a clean install of macOS without migrating or restoring from a backup has the same problems, then you have some sort of hardware issue.

You can check the health of the hard drive using Knoppix Linux. Use the Knoppix .iso file as a source for Etcher (Mac/Windows/Linux) which will create a bootable Knoppix USB drive. Option Boot the Knoppix USB drive and select the orange icon labeled "EFI". While Knoppix is booting the computer may appear frozen on the boot picker menu so give Knoppix lots of time to finish booting.

Once Knoppix reaches the desktop click on the "Start" menu icon on the lower left corner of the Taskbar and navigate the menus to "System Tools ---> GSmartControl". Within the GSmartControl app double-click on the laptop's hard drive icon to access the drive's health information. Post the full report here. If you have a Fusion Drive setup (or multiple internal drives), then post the full report for each drive.

If you had this computer repaired by Apple or an AASP within the last 90 days, then take it back to them so they can make things right while you still have a part warranty on the previously replaced part(s).

Well, you've omitted plenty of essential information. Here are just a few examples:

• What is the exact year and model of the MacBook Pro?
• What is its boot device exactly?
• How is the boot device formatted?
• What operating system are you attempting to boot?
• How was it loaded?
• What disk or media is your Time Machine information stored on?

Your answer number 2 sounds impossible. You cannot both boot from a device and use that device for backup storage.

How to create a bootable installer for macOS - Apple Support

Yes.