I just setup a new computer that's been setup with the local admin that we want on all Macs but the standard user that gets created through JAMF by asking for user verification during out of the box setup does not show up as an option for sharing. Can this not be granted to non-admin users instead of only seeing "Administrators" or local admins?
In the screenshot "SysOp" is our local admin user for techs to use. My user is not seen here which is a standard user.
Sorry if I am missing why you want to include a standard user, so my answer is based on my assumptions. You can add a standard user, but given that the listed accounts are used by the person on the other end, not sharing the computer, the person remoting in would need the credentials of the account listed, so it makes sense to have admins accounts listed. If the standard user is the device user, the person remoting in would need to know that users password. That's not really what you want.
One last thing, if you are using this Screen Sharing so users can share their screen, don't. Any App like Zoom, Teams, will add Screen Sharing to the Security and Privacy/Privacy/Screen Recording section (which the device user needs to tick.
I would first like to say. If you are using Screen Sharing in an enterprise environment, don't. It is an extremely consumer focused tool, so much so that Apple does not even use it for remote support. If you are going to use Screen Sharing, I recommend setting it up through Remote Management so you get some more options and control. Apple uses Beyond Trust (formerly bomgar) which may be worth a look.
As far as your account not displaying. How was it created? I am going to take a guess and say the account you are wanting to see does not have the correct group membership. Also try this on a different device just to rule out macOS being dumb.