A family of Microsoft relational database management systems designed for ease of use.
There are two issues here:
1. The parent form is presumably based on an Invoices table or similar, in which the customer is a foreign key. If you are really using the customer name as a foreign key of text data type, personal names can legitimately be duplicated so are unsuitable as keys. However, if you have used the lookup field wizard when designing the table, while you see the customer name in the table and in the parent form, the field will in fact almost certainly be a long integer number data type referencing the primary key of a referenced Customers table or similar.
2. The subform's table will, I assume, be based on an InvoiceDetails table or similar. This table should contain an InvoiceID foreign key column, but must not contain a Customer column. A Customer column would not be transitively dependant on the primary key of the table, which would therefore not be normalized to Third Normal Form, and consequently be open to update anomalies such as that which you are experiencing when selecting a different customer in the parent form. There is no need to include the customer name or invoice date in the subform's table as each row in that table maps to a row in the parent Invoices table via the InvoiceID keys. In other contexts than the form/subform, e.g. in a report you would simply need to join the tables on InvoiceID in a query used as the report's RecordSource.
For an example of an invoices form you might like to take a look at InvoicePDF.zip in my public databases folder at:
https://onedrive.live.com/?cid=44CC60D7FEA42912&id=44CC60D7FEA42912!169
Note that if you are using an earlier version of Access you might find that the colour of some form objects such as buttons shows incorrectly and you will need to amend the form design accordingly.
If you have difficulty opening the link, copy the link (NB, not the link location) and paste it into your browser's address bar.
This little demo file is primarily intended to illustrate how to output a report as a PDF file, but happens to use as its example a simple invoicing application.