Currently, there's no option to create Customer and Supplier with the same name under a single ledger. This is because the system will not allow you to enter the same information on the Customer and Supplier details. You'll need to add extra characters on either name to save the information.
For now, there isn't a way to assign a ledger account in the customer view. I suggest to manually choose the item that's associated with the income account for the customer transactions.
For the Supplier, you can assign a single expense ledger account only. Just choose the Default expense account in the Supplier Information window. View the screenshot below for your reference.
I also recommend running the Account QuickReport of the ledger account you use for your supplier. It helps you view all its associated transactions within a specific period. Just click Run report under the Action column in the Chart of Accounts menu.
You can visit our resource hub below to learn more about effectively managing your suppliers and your customer list.