From a clarification under the question, when you go to a supermaket, the person you pay for your groceries is the cashier:
: one that has charge of money: such as
a : a high officer in a bank or trust company responsible for moneys received and expended
b : one who collects and records payments
c : an employee (as in a store) who handles monetary transactions
It's the third sense here that's the most relevant with respect to a supermarket.
In supermarkets, there can be 10 or more cashiers, each in their own lane.
This position might also be referred to as a clerk (or checkout clerk):
3 b : one employed to keep records or accounts or to perform general office work
// a bank clerk
3 c : one who works at a sales or service counter
// a store clerk
// a grocery clerk
In my personal experience, I've heard clerk (or checkout clerk) used more often in UK English. However, a comment under this answer indicated that hasn't been the experience of somebody else, at least not when it comes to regular speech.
With respect to a different answer, at least in North America, those 10 people would not be called a grocer or shopkeeper. Those words, are reserved for the owners of the stores. In a small store, it could be that the owner is also serving in the same role as cashier, but it's almost certainly not the case in any large chain store with more than a single checkout location.
Not to belabour this point, but a supermarket is a large store with many different checkout aisles and cashiers. It would be highly unusual (and it's never happened to me) to ever see the owner of a supermarket working the cash.
Small stores, which have only a single person on the cash, are not called supermarkets. They are called corner stores or variety stores. Only with these informally so called mom and pop stores would you be likely to see the owner (grocer or shopkeeper) at the cash.
To address another comment under the question, the machine itself that's used to store the money is not called a cashier (who is a person). It's called a cash register:
: a business machine that usually has a money drawer, indicates the amount of each sale, and records the amount of money received
// Employee worked at the cash register then put on gloves and assembled plates of food without washing hands between tasks.
— Ebony Day, azcentral, "Too warm coleslaw among violations found in restaurant inspections," 15 Apr. 2020
So, cashiers work at cash registers.