Can't initialize the SSD installed in place of the optical drive on a MBP 15 late 2008 running El Capitan

Let me suggest you try this:

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility from the Utility Menu and click on the Continue button.
3. When Disk Utility loads select the target SSD (out-dented entry w/type and size info) from the Device list.
4. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
5. Set the partition scheme to GUID.
6. Set the Format type to APFS (SSDs only) or Mac OS Extended (Journaled.)
7. Click on the Apply button, then wait for the Done button to activate and click on it.
8. Quit Disk Utility and return to the Utility Menu.

Select Restart from the Apple menu.

Move the SSD to where the original HDD is located. Move the HDD to where the optical drive was connected. You may need some hardware for this. Go to OWC and look at their DataDoubler product.

Be aware that you will need to be able to boot the computer from the HDD using Option Booting so you can format the SSD before installing macOS on it. Not until then will you be able to boot from the SSD. Alternatively, you can create a USB flash drive installer to format and install macOS on the SSD.

OK. First thing that comes to mind is that the Kingston device isn't working or isn't compatible. You might try exchanging it for a different make. I've always had good luck with SSDs from OWC and OCZ products. The second thing that comes to mind is that the SSD is defective. The third thing that comes to mind is the cable connection. If there is a cable between the drive and the computer, then try replacing the cable. If the connection is direct without a cable, then check that the connection is correctly mated. Being off by one set of pins is an easy mistake, and if that's the case then the drive won't work.

The last thing that comes to mind is how the SSD was formatted when you received it. If it was formatted using NTFS, then that causes problems for macOS because it can only read the drive. That means it is hard to format on a Mac. The work-around is to try formatting the drive using MSDOS. If that works, then the drive is writeable. You can now format it for macOS - Partition scheme: GUID; Format: APFS. You may have to try several times using Disk Utility to get the change from NTFS to MSDOS.

To know whether the SSD itself is OK, set it up in an external enclosure, adapter, or "toaster"-style device.

Sometimes MacOS is reluctant to erase a device that has data on it already. The way around non-MacOs devices with data already present is just as Kappy suggested.

For a drive does not contain data, anecdote suggests, if you ask it to erase, and when refused, immediately ask again, it is sometimes successful.

Your first screenshot shows "Partition Map: Not Supported". Did you make sure Disk Utility erased the SSD as GUID partition and MacOS Extended (Journaled)?

Also the SATA Controller for the Optical Bay in these older Macs are known to have issues with SSDs. Apple only ever expected the Optical Bay to contain a slow optical drive and not an SSD which transfers so much more data at much higher rates. The controller and the cable cannot always handle an SSD reliably. I know Apple did update the system firmware on some later MBPros to fix some of the issues. I've also heard of people wrapping the flex cable to shield it to minimize issues with the higher transfer rates of the SSD, but this must be done carefully so not to cause an electrical short.

Plus if your Kingston SSD is a SATA III version which is most likely, it could have trouble auto-negotiating the SATA link speed in the optical bay (or even on the main hard drive controller). I know OWC still sells a 3G (aka SATA II) SSD just for these reasons.