Signatures in Apple Mail

Try this:

1. Leave Mail.
2. Open System Settings.
3. Click your name (top left).
4. Click iCloud.
5. Click "iCloud Drive".
6. Click "Apps synchronizing with iCloud Drive".
7. Disable Mail.
8. Click Done.

Once this is done, open Mail and create your signatures. Quit and reopen Mail to check that the signatures are still present. If so, restart your Mac and check for the presence of your signatures again.

If everything seems normal, resume steps 1 to 8 but, this time, activate Mail in step 7.

Perhaps, if you walk us through the precise steps you have taken, we maybe able to spot where it issue is ?

FYI - on 3 different machines I own and operate all running Ventura 13.4.1 - have zero issues adding or remove Signatures

Saying this does not work and that does not work is fine.

But what have you done outside of rejecting all suggestions put forth

We want to help solve this issue.

Not knowing exactly what actions you have already attempted seem fruitless and pointless to keeping offering.

Perhaps the Apple Support Experts are willing to listen.

Get Support Choose a product and we’ll find you the best solution.Start now and open an Apple Support Ticket as they are Apple Employees to deal will these types of issues . 

Have re-read the original question " Cannot save/modify or delete a signature. ".

Is the e-mail account used to perform these actions associate with an e-mail Account with a Third Party e-mail Service ?

We, in the collective sense of we, are issuing instructions for using iCloud as the e-mail provider

In theory, it should work some but not all e-mail providers

To create a signature in macOS 13.4.1 follow the instruction from this Apple Support article which contain all the details

Create and use email signatures in Mail on Mac

Excerpt from above link

You can create prepared text, called a signature, to include when you send email messages. For example, you can create one signature for your work email and another signature for your personal email.