Reinstall through Internet Recovery

Looking back at my earlier post, I forgot to mention that you can eliminate the internal hard drive cable as the issue by removing the SSD and connecting it externally using a USB to SATA Adapter, dock, or enclosure.

I also just noticed you were erasing the volume and not the physical drive. In Disk Utility just above the left pane, click on the "View" drop down list and select "Show All Devices". The physical drive should appear in the left pane of Disk Utility. Select the physical drive and erase it.

Good luck.

rcosta887's suggestion is a very good idea to try. If it doesn't work, I would suggest performing an ATA Secure Erase on the SSD which will reset it to factory defaults just in case it got into some weird state that rcosta887's suggestion won't work. I have had to zero out a USB stick several times because Disk Utility could not partition/format it correctly otherwise. To perform an ATA Secure Erase, download the free version of Parted Magic from this link here and create a bootable USB drive using Etcher. Option Boot the Parted Magic USB drive and select the orange icon labeled "EFI". There should be an erase option on the Parted Magic desktop. Make sure to select the "ATA Secure Erase" option and not the other slower option to "zero" the drive. The ATA Secure Erase is a hardware feature built into most SSDs and will take 30-60 seconds when activated.

It is possible you have a bad hard drive cable which is very common in these laptops especially when upgrading to an SSD.

Also try an SMC reset and PRAM reset (hold the PRAM reset for at least three chimes).

What I would try is to actually repartition it (on your desired MacBook Pro if possible, otherwise on the other Mac which seems to work better with it) and change its partition type temporarily to a Windows MBR (Master Boot Record) partition. Then go ahead change its partition type to the (required for Mac) GUID partition type.

By doing this it will flush out the boot table sector on the hard drive without needing to use any low-level commands and there's a good 50% chance this will work to get you back on track.

I would suggest replacing the hard drive cable in the laptop. While the OS may install I expect you will begin to have problems once you start booting directly to the SSD since you will be reading & writing to the SSD at the same time unlike the install process where you are mainly writing to the SSD.

The "missing firmware" issue is due to Apple being stupid. Mojave requires a hidden EFI partition to exist on the internal drive or Mojave won't install. The hidden EFI partition is a boot partition containing a boot loader and it is also used to store a firmware update so it can be applied on the next reboot. I have no idea why the hidden EFI partition on an external drive cannot be used, but Apple has decided to make life difficult for us. I wish their error message would be more explicit like "macOS requires the internal drive to contain a hidden EFI partition to hold required firmware updates even when installing macOS to an external drive". One way around it is to install your old drive (or any Mac formatted drive) internally until after the macOS install completes.