You need to hack a little bit to achieve this. First of all, don’t touch WordPress default login. That can mess things up a lot.
I think for your case, it would be best to create a secondary login system. For this, you would need to store the customer numbers in a separate table.(It’s not apparent from your question how sensitive this customer number is, how they are generated and their effect in contrast to the password. So store them appropriately.)
Create a page template for the login page.(I’m assuming the customer number is added by admin, in which case you need to create a admin page in the backend to add/edit/delete customer number)
Now you need to store the login details in a session. But WordPress doesn’t allow session. You can activate PHP Session or use this plugin instead: https://wordpress.org/plugins/wp-session-manager/
Now on the page where you want to restrict access, you can check whether user is logged in and show the content or send them to the secondary login page.
If this is needed only a single page, you might not need session at all. When user visit the page, show them a form to submit their customer number and on form post do the validation. This will require user to input their customer number in every page load.