Password doesn't work for software update but does for everything else
This will be a little tricky to pick apart remotely. Your edits are super helpful to come up with next steps.
I would use the command line and try this in the terminal app.
sudo softwareupdate --all --install

You will be prompted to enter your password and if that accepts your password and starts to download and install the updates, you know it’s just a GUI account issue.
If you don’t like the command line, another path is to make a second admin user and log out of the problem user (Apple menu - log out) and then try the update from the second account. Either way, you can reduce the potential causes and maybe get your update done, too.
I get the same prompt, but it works for me today when yesterday I got password errors and I changed nothing. I have Apple Silicon on my Big Sur mini so ymmv.