Credit Card is a Revocable mode of payment. Cardholder buys something from a retail store on Day 1. They get the credit card statement on Day 30–45. When the credit cardholder sees the debit entry on their statement pertaining to this store, they can dispute the charge on various grounds including but not limited to “I never bought anything from this store”. Many times, cardholders simply forget their shopping visit or the store name on their credit card statement is different from the store they remember having visited. Nevertheless, when they dispute a charge, the Credit Card Issuer Bank reverses the charge and raises a “Chargeback” ticket on the Merchant i.e. Retail Store. It then becomes incumbent on the store to prove that the cardholder had indeed shopped there. Failure to do so would mean clawback of the money already received by the store from the bank and fees on top of that.
To fight any future chargebacks, some stores take wet ink signature on the chargeslip as proof. Many don’t. But they at least collect the cardholder’s phone number, so that they call the cardholder and jog their memory about having visited the store on so-and-so date and time and bought so-and-so item. Many times, cardholders recall the visit and purchase after this call and agree to withdraw their disputed entry. This will lead to closure of the chargeback ticket, which means the store gets to keep the money for the goods it sold and will not suffer any chargeback fees.
Cash is an Irrevocable mode of payment. Once paid it’s paid. Customer can’t demand it back. Therefore, store does not need to seek any information about the Customer.
That said, retail stores may still ask for telephone number of customers paying with cash in order to enroll them into their loyalty management / customer engagement programs. This happens quite regularly in markets where people are not very privacy-conscious but may backfire in other markets where people don’t like to share their PII with anyone and everyone.