Changed user name, but old name appears under "Sharing & Permissions"

The computer name in Sharing has nothing do do with a user account name except the Startup Assistant uses that name to create the first computer name. Changing one or the other has no effect on the other.

Changing your user account name will also not change the ownership of the files in the account.

Files within your home should be owned by your user, there shouldn't be any Group associated with the files, and everyone should be no access.

You may have to resort to this: Resolve issues caused by changing the permissions of items in your home folder - Apple Support

To find out if it's system wide or user specific, try this...

Open System Preferences>Users & Groups, unlock the lock, click on the little plus icon, make a new admin account, log out & into the new account.

Does it work in the new account?

So, just to make sure I understand the "problem" you're trying to resolve:

In the "Sharing" System Preferences, the name listed in the "Users" column is the "iMac" name and not the "mini" name? ]

I think this is really just a cosmetic thing, and not a real problem. I am guessing that when you had sharing first setup (when you were on the iMac) that the iMac user name got saved in a sharing preference plist file, rather than a userid number. Since the way you did the username change didn't really change the userid value, that's why there is no real problem. This is just a guess, as only Apple can likely explain the "why" in this case.

However, if you would like things to reflect the real state of your Mini system now, you could remove the old "iMac" name from the list of users in Sharing and then add the "Mini" username. Just highlight the "iMac" username and click the minus icon below and then click the "plus" icon to add the "Mini" username

Good luck...

Does the Get Info allow you to add the new you?

To see if the new account shows proper Get Info info for the new user, & if your name is offered as a choice in adding rights.

So, your Home Folder says iMac beside it?

Are you sure the Mac wasn't an "open box" or otherwise used previously?

Otherwise, I don't know why it would have a different name.

When you migrated, was there an account already setup on the Mini?

Do any other accounts show in Sys Prefs>Users & Groups?

Just remembered, MA only asks for a new name to migrate to if that username is already in use on the new one... I guess we'll see what MA thinks the username is anyway!