How do I get and install the update for Safari 16.3.1 for macOS Monterey?

You don't! It's a new build of 16.3. There IS no 16.3.1.

16.3.1 is just an illusion created by misinformation on an apple security webpage.

The website About the security content of Safari 16.3.1 - Apple Support should be referring to it as 16.3 not 16.3.1. I wrote more about it here:

Shouldn't the Feb 13, 2023 Safari for Big Sur and Monterey be version 16.3 not 16.3.1?

You don't! It's a new build of 16.3. There IS no 16.3.1.

16.3.1 is just an illusion created by misinformation on an apple security webpage.

The website About the security content of Safari 16.3.1 - Apple Support should be referring to it as 16.3 not 16.3.1. I wrote more about it here:

Shouldn't the Feb 13, 2023 Safari for Big Sur and Monterey be version 16.3 not 16.3.1?

Less

As someone points out here and on your other thread, this is no longer strictly true. There IS a version 16.3.1 NOW ... but it's ONLY for Big Sur. This is such a stupid problem to have and it was solved decades ago with version numbers that should always increment on a new release. Changing a build number doesn't let my JAMF instance detect which clients haven't updated because there are now 2 different versions both named 16.3.

Seems to be a common affliction: Updating Safari 16.3.1 yields v16.3 - Apple Community

You still don't.

Safari 16.3.1 is a thing now but it is exclusively a Big Sur 11.7.4 thing.

And they finally fixed the webpage to change it back to 16.3.

On Feb 15, 2023 they came out with Big Sur 11.7.4 and Safari 16.3.1 and you can only get Safari 16.3.1 if you have installed Big Sur 11.7.4