How does one checks which customer made a payment?
Do you really care which customer paid, or do you care which bill was paid?

Say you have customers Anna and Bob, who both order something. Bob notices he's out of money, so Anna says she'll pay for both. If you really want the sender to reveal their identity, your system would already fail, as Bob never pays anything.

In almost all cases, all you care about is which bill is paid, and you have no need for the payer's identities (which could be a privacy violation in some cases).

The normal way to do this in practice is by creating a new payment address for each payment you want to receive. You track which bill was paid by tracking at which addresses you received money. Creating new addresses is very cheap, and they are sufficiently long that it is exceedingly unlikely that the same one will ever be produced twice.