Hypothetically speaking, if A shoplifted and got caught , no statement was signed and no police report was filed. In short, no authority was involved. But a photo of A and A's identity card was taken. A paid a sum of 3 times the value of the goods stolen. Can the shop sue A even they made an oral promise not to? Can they file a police report afterward? This happened in Malaysia.

Yes. Some stores do want shoplifters to have a record, and restitution doesn't wipe it out. Presumably A has proof of payment made.
But it is something A is just going to have to worry about.
Each store has a different policy.

Since there was no written agreement, they are free to do anything they want, They can pretend they never got the money, if there was no written agreement, and/or no reciepts of payments

Since there was no written agreement, they are free to do anything they want, They can pretend they never got the money, if there was no written agreement, and/or no reciepts of payments
No receipt was issued but A paid with her credit card hence there is a transaction record .