SSD not showing up in Utility

If you have installed a third party NVMe SSD, then you need to boot from macOS 10.13+ in order to see the SSD. Your screenshot shows you have macOS 10.13 installed, but I'm guessing when you boot into Recovery Mode you are booting into macOS 10.9 which originally shipped with the Mac Pro. In this case you will need to create a bootable macOS 10.13+ USB installer using the instructions in this Apple article:

How to create a bootable installer for macOS - Apple Support

FYI, within Disk Utility you will need to click "View" and select "Show All Devices" in order to see the physical drive on the left pane of Disk Utility (macOS 10.13+).

If you received the Mac and it was booting to macOS on the internal drive, then yes, you can use your 2011 Mac to create a bootable macOS 10.13 USB installer to at least get the Mac Pro booting to the internal SSD where you can at least upgrade macOS to 11.x Big Sur.

How to create a bootable installer for macOS - Apple Support

If the Mac Pro was not booting macOS from the internal drive when you received it, then you won't be able to install macOS to the internal drive without an original Apple SSD.

In order to use the hack to upgrade the system firmware for Monterey, the Mac Pro's system firmware will need to be at least 195.x.x.x.x or greater and you will need to have macOS 10.13+ installed on the internal drive.

The base system is just a disk image on an didk/SSD. It is not the entire SSD as you are aware.

Yu said in your initial post that:

"I have a 2013 Mac Pro SSD is not showing up in the utility list or in the terminal, but does show in about this Mac. Only the base boot drive shows up in the utility list and terminal. Any ideas on how to get the computer to read the SSD? Just installed a new SSD from OWC with same issue."

In the first sentence what SSD was installed internally? An Apple SSD?

Does your internal SSD use an adapter or does it directly go into the internal slot?

What disk are you booting (startup disk) from? The internal disk?

With the OWC SSD have you contacted OWC? They have excellent support.

The 2013 Mac Pro can run Monterey. That is what I am using now.

My startup disk is an external SSD since I have not got around to reinstalling the non-Apple SSD internally since I had to put the original Apple SSD back in to upgrade to Monterey.

Do you have an Apple SSD that installs internally?

Do you have an external USB (or thunderbolt 2) SSD?

DO you have access to another Mac or does you Mac work?

You can install install Big Sur on an external SSD using a Nog Sur installer. That may upgrade the firmware to use an NVMe drive.

What is firmware (or EFI ) version is installed on the Mac?

About this Mac>Overview>System Report> Hardware

If you removed the internal SSD card and replaced it with one from OWC, your display is as expected.

If you did something else, please explain. Is that new SSD in an external enclosure? what interface to the computer?

What does Disk Utility say about it?

The Disk Utility GUI is basically a graphical wrapper around "diskutil" and several other command line utilities. When using the command line it is best to filter for just the "internal" drives since you will get a list of over twenty virtual drives/volumes from the macOS installer ( "internal" may not be an available option with older versions of macOS -- should be ok for 10.10+).

diskutil  list  internal

As far a seeing the physical drive.....If the Mac doesn't have at least the system firmware provided by the macOS 10.13+ installer, then a third party NVMe SSD (aka "blade" SSDs) will not be seen at all. macOS 10.13+ added support for NVMe SSDs with a driver and also updated system firmware. Both the updated system firmware & NVMe driver are necessary to see a third party (or even Apple) NVMe SSD. The OWC Aura SSDs are NVMe based SSDs. OWC mentions the macOS 10.13+ requirement on the product page of the website.

If you replaced the original hard drive with a third party 2.5" SATA SSD (such as the OWC Mercury Electra series), then it should not matter what OS you are installing. If you are using a 2.5" SATA SSD and you don't see the 2.5" SSD, then maybe you have a loose cable or a bad SSD. You can connect a 2.5" SATA SSD externally using a USB to SATA Adapter, drive dock, or enclosure to test if the SSD is working.

Also note that to install Monterey you need an Apple SSD installed since the Monterey installer includes an update to the Mac's firmware and the installers checks to see if there is an Apple SSD installed since it writes to the SSD in order to update the Mac's firmware. If not Apple SSD the Monterey install terminates.

Once you install Monterey, you can install a non-Apple SSD and install and run Monterey on the non-Apple SSD wit no problem.

The "MacOS Base system" with under 2GB is Internet Recovery. It is in the computer's Read-Only Memory, and can not be used for anything else.

A "MacOS Base System" of size above 5GB is likely an Installer Disk image, also not usable for anything (except as a software Source for installing, not as place to install.)

It could be the SSD is a standard NVMe drive.

For the 2013 MP to use and internal NVMe drive it firmware would have to be updated to support that:

 The MP61.0120.B00 boot ROM included support for NVMe booting (found in the High Sierra update).

It could be that your Mac does not have the required firmware update to use an NVMe SSD.

What was the highest macOS version installed and working on the Mac?

My 2013 MP uses an adopter wth a standard NVMe drive.

MCMac_Pro wrote:

I have Big Sur installed. Monterey is for M1 chips not Intel.

No, macOS 12.x Monterey is compatible with Intel chips.

Big Sur 11.6.1 is highest I am able to run.

Perhaps if you are using a third party NVMe SSD. You can update the Mac's system firmware by installing an original Apple OEM SSD into this Mac and install macOS 12.x Monterey in order to update the system firmware. I believe @lllaass suggested another way to upgrade the system firmware if you don't have the original Apple OEM SSD.

If let’s say the PCI SSD in the Mac is good that came with the with the computer when I bought it(2nd hand). How would I do a factory reset if the Mac HD is not populating as a location to reinstall the OS?

You are using a third party NVMe SSD which requires macOS 10.13+ in order to see & use the SSD! This is why the "diskutil list internal" command gave you no results since you were most likely booted from macOS 10.9 which is the version of macOS which originally shipped with this Mac and may be the only online installer you are able to access.

You need to be installing macOS 10.13, 10.14, 10.15, or 11.x. Currently these are your only options for installing macOS to the third party NVMe SSD. You can NOT use or install macOS 10.9 to 10.12 to the internal third party NVMe SSD.

You can install macOS 10.9 to 10.12 to an external drive in order to upgrade the external drive to macOS 10.13+ so you will see the internal third party NMVe SSD, or you can then create a bootable macOS 10.13+ USB installer so that you can finally boot the correct installer which will see the internal third party NVMe SSD.

MCMac_Pro wrote:

The PCI non apple internal SSD has an adapter. I do have an external HD to use. So I should be able install Monterey on there to update the firmware on the computer to be able to run Monterey?

No, because the macOS 12.x Monterey installer will refuse to update the system firmware while a third party SSD is installed internally even though you are installing macOS to an external drive. You need to install an original Apple OEM SSD internally, then you will be able to install macOS 12.x Monterey and update the system firmware. After the system firmware has been updated, then you can reinstall the third party NVMe SSD and install macOS 12.x Monterey to the internal or external drive.

I did stumble across a method that may allow updating the system firmware with a non-Apple SSD installed internally, but I am not posting the link here as I don't have faith that you even understand the situation here, much less using the command line. Besides it may require booting to macOS 10.13+ in order to attempt the hack which so far you don't seem to understand.

This whole can of worms came just trying to factory reset the computer…

A computer which was modified with third party hardware. This is one of the dangers of purchasing used computers, especially a used Mac.