FYI, for reference here is the other thread where we've been trying to assist....it contains a few posts that may be relevant here:
Problems with installing macOS Sonoma 14.… - Apple Community
In that other thread (especially your last couple posts) I was beginning to suspect you are confusing things with your various external disks. I think you are mixing up the installer USB disk and a full macOS USB installation. This seems to be born out by the pictures you have posted here although you don't inform us how you were booted to the installer when they were taken. Your picture seems to show you trying to install macOS onto the USB installer.....that will fail unless you first created a new partition containing an APFS volume next to the installer partition. The second picture is confusing since it shows macOS being installed onto a volume named "OS", but that doesn't match the first picture which shows the highlighted destination volume as "Install ma...S Sonoma" which I take to mean is actually "Install macOS Sonoma" which is your macOS USB installer.
You need to use a USB stick for the macOS USB installer. You do not want to use a USB SSD drive for the bootable macOS installer. Your Kingston SSD should only be used for the full installation of macOS.
Use a USB stick for the bootable macOS USB installer. Follow the directions in the following Apple article where it mentions you need to erase the USB stick as GUID partition and MacOS Extended (Journaled) with a name of "MyVolume" (notice there are no spaces in the volume name).
Create a bootable installer for macOS - Apple Support
You also need to prepare the new Kingston SSD as well. You need to erase the whole physical Kingston SSD as GUID partition and APFS (top option). Make sure to give it a unique name. While the default macOS volume name of the internal drive from the factory is "Macintosh HD", you don't want to use that since it will just cause confusion, so you may want to use "Macintosh SSD" instead. Within Disk Utility you may need to click "View" and select "Show All Devices" before the physical drives appear on the left pane of Disk Utility. It is important that you select the physical drive to erase since some external drives ship from the factory with odd partition layouts and even hidden items that can negatively affect macOS. Here is an Apple article showing how to erase the entire physical drive:
Erase and reformat a storage device in Disk Utility on Mac - Apple Support
When you boot from the USB installer stick, you will want to select the external Kingston SSD named "Macintosh SSD" as the destination location assuming you go with my naming suggestion.
If you do this, then you should have no problems assuming that the hardware is all healthy & compatible.
FYI, please make sure to be specific on the steps you perform and any errors encountered, and make sure we know which step and/or system configuration is being used when attaching any pictures. And please make sure to clearly state whether you mean the "installer" or the "full macOS installation" when discussing the contents or use of any drives.
Good luck.
