Livepatch is turned on
Note the following line in the second screenshot in the question:
patch state: ✓ no livepatches needed for this kernel yet
Also note the following lines in the third screenshot:
NOTICES
The current kernel (5.19.0-43-generic, amd64) is not supported by livepatch.
Supported kernels are listed here: https://ubuntu.com/security/livepatch/docs/kernels
Either switch to a supported kernel or `pro disable livepatch` to dismiss this warning.
These lines indicate Livepatch is enabled but is not available for the installed kernels.
The rest of the answer tries to clarify some of the confusion.
You don't need livepatch
What is a kernel? How is it updated?
A kernel is the most fundamental part of an operating system like Ubuntu. It starts up very early when the computer starts and it controls everything.
From time to time the kernel is updated with security fixes. This is part of normal software updates. You can setup security updates to update automatically using the Softwares & Updates app. Or you can manually update your Ubuntu using Software Updater app. You may also use the following commands to keep the computer updated:
sudo apt update
sudo apt full-upgrade
Note, you will need both the commands in that order.
When normal software such as Libreoffice or Firefox is updated Ubuntu does not need to be rebooted to use that software. You can close LibreOffice and start it again and the updated version will start.
This does not normally work for kernel updates. When you use the app Software Updater and the kernel is updated you will see a notice that says the computer must be restarted for the kernel update to take effect.
In summary:
- Keep your computer updated and kernels will be updated as and when needed.
- Restart your computer when prompted to make sure you are using the most recent kernel.
What is kernel livepatch?
Kernel livepach is a mechanism that allows kernels to be updated without restarting the computer. This is like changing a tire of a car while driving it. This is needed in computers used in critical situations. For example, if the computer controls a nuclear power plant, rebooting the computer may not be always possible. My guess is your computer does not control something as critical and you should be able to reboot your computer as and when needed. Thus, you don't need livepatch.
But what if you really really need livepach?
Livepatch is not yet available for Ubuntu LTS HWE kernels. The list of the kernels that are supported by livepatch is availalbe here.
You have two options:
First, use the non-HWE LTS kernel. For Ubuntu 22.04 LTS it is the kernel was released with, version 5.15. Since you are already using kernel 5.19 the current HWE kernel, I won't recommend that.
See How to go back from HWE (5.13) to GA (5.4) kernel in Ubuntu 20.04 LTS? for how to go back from HWE to original kernel. The process is similar for Ubuntu 22.04, but it may not work with most recent hardware that needs the newer kernel.
Second, wait a few months when the Ubuntu 22.04.3 comes out. That will come with a newer version of the kernel and it will support livepatch in HWE LTE kernels.
See this question and answer for more: Livepatch -- plans to support official HWE kernels?
See this answer for the differences between updating Ubuntu and Windows work: Does Ubuntu force updates like Windows or I can choose if I want to update?
Hope this helps