Yeah, I have this issue too. In my scenario, I have an HTML code signature created using the process outlined here: https://www.daretothink.co.uk/html-email-signature-in-apple-mail/
I've been using signatures created from those instructions for almost 10 years. I recall that I had to do something different to get it to work in iOS when emailing from my phone but I think it was a simple as copying & pasting the signature on my phone from a sent message. I say 'recall' because that has continued to work without issue since I did it the first time and survives through iOS upgrades and new devices without fail. Huzzah and kudos there, Apple. You'd think that the process would be that simple on desktop OS, but it's not. Seriously don't understand how Apple hasn't figured that one out yet... Boo. Hiss.
And syncing? Pffft, THAT is, as far as I can tell, the root of the problem that I still haven't figured out how to work around.
Everything was fine as signatures go until I put another Mac into service. The moment I set up signatures on the new Mac Studio, the ones on my MacBook Pro stopped working. The named signature was still present, but it reverted to the generic "Name, sent from my ______" nomenclature that the OS plugs in by default when you create a new signature. And when I fixed it on my MacBook Pro using the same process, the signature on my Mac Studio was overwritten to the default text. I've gone back forth until I decided I just couldn't have a proper signature on one of devices. How very Windows 95 of you Apple.
When you create an HTML signature, you have to use TextEdit to put the info into an existing signature (again, why Apple????) but if you don't lock the document, the moment you launch Mail it will overwrite whatever you put in there with that default text. I have a sneaking feeling that the OS is doing some SUDO black magic when syncing accounts that allows it to overwrite whatever signature you created regardless of whether it's locked or not. But why? Who knows...
This whole process makes is seem like Apple has no idea how professionals use their products. But that's nothing new (#finalcutprodevteam).