A new membership isn’t the same things as a new account. “Memberships” are properties of the accounts. For example, a scout could have a single account with a membership in both a troop and a crew, and that would be perfectly normal. It doesn’t work seamlessly (yet!), but it does work. The new troop membership would work the same way.
To add a membership in your troop, the parent can go to:
My Dashboard -> My Family -> Scout’s Name -> Scout’s Memberships -> Add (red button at the top)
I am concerned that the information from the scout’s account isn’t allowing you to transfer him in, though. If the information matches, it should work.
Keep in mind that no matter how the new membership in your unit gets added, the scout still needs to turn in the appropriate “paperwork” to council for the transfer. Make sure it has the correct BSA ID on it and is marked as a transfer application. You probably already know this part, but it has caused us no end of trouble when we forget, so I try to remember to include it in these types of conversations.
If the birthdate is wrong in Scoutbook, you probably need to talk to your council registrar to get that fixed in the system. I think that if they fix it on their end it will push out to Scoutbook, but I’m not 100% sure on that. If it doesn’t match, it could cause issues down the road where a new BSA ID/account gets created instead of properly linking to the existing ones.