Migration Assistant din't move the Home folder to tthe admin account. Why not?

This page tells how to do the migration. https://support.apple.com/en-us/HT204350

Note that it covers the case where you have a user already created on the target computer with the same name as on the previous computer. I think this is what you did. Apparently you responded to the prompts in such a way that a new separate user was created. It is actually much easier to migrate on the very first boot up before any users are created on the new computer. One of the users on the new computer should have everything that was in the older computer account. But sometimes the account is renamed unless you instruct it to overwrite the one you created. See below.

I can't tell what has actually happened here. I'm not sure I'm comfortable giving more suggestions about how to resolve this without inspecting the folders, which I obviously cannot do. I don't understand the "user folder not a home folder," I would have expected this to be for a user that you could log in as. Have you tried to do that?

Your mistake was creating a user account with the same name on the MBA. Migration Assistant won't be able to migrate your old user account because it doesn't expect there to be an existing account with the same name.

The best approach would have been to use Setup Assistant, when you were prompted, when you turned on the MacBook Air for the first time, too late now.

Now, your best approach would be to create a new Admin account on the MacBook Air, with a name other than the one you want to migrate, i.e. "temp." Sign-out of your duplicate account and into the new Admin account, then remove/delete the account that you initially created on the MacBook Air. Then you can use Migration Assistant to migrate your user account over from the MacBook Pro. Once the migration has completed you can then sign-out of the new Admin (temp) account and login to your migrated user account. If the migration is to your liking, you can then remove the 2nd admin account.

Remember, Migration Assistant is for copying an existing user account to another Mac that doesn't already have that user account.

It's Lanny, not Larry.

Once again, you should create a new admin account on the MacBook Air, named temp. Then you sign in to that account so that you can delete/remove the previous account that you made on the MacBook Air. Once you've done that, you can use Migration Assistant to add your account from the MacBook Pro to the MacBook Air. If the Migration Assistant brings your prior account to the MacBook Air, you will then be able to use that account on the MacBook Air. Then it will be safe to delete/remove the temp account.

Using the "User & Groups," in System Preferences, you're going to remove/delete the user that you created that has the same name used on your MacBook Pro. This can only be done if you're signed in as the temp user admin account. Don't use the Finder for any of this. Leave the Applications and Library folders alone.

When you use Migration Assistant, you can then exclude the applications portion.

No, there is no renaming of accounts involved. You are not migrating anything to the temp account. The temp account is to only be used to delete the existing user account on your MacBook Air because you shouldn't have created one there to begin with.

If you don't understand this, boot to your Recovery partition and erase the drive and start over from the beginning.