Not long ago (in dog years) you could look at online purchase history in Apple and see products along with their serial numbers. Some items appear in my products that are used in conjunction with iCloud and my AppleID online; even some second hand. And older co-use products could have retained serial numbers in associated product Apple software; I see this in some of mine.
This kind of information had been available from a sales/support sign-in page; possibly where you may also check in applecare, etc; but details have not been retained to the degree they once had.
Not sure when you bought this product or if you have looked for any residual information where it may/have appeared. Older purchased items long out of initial warranty and w/o applecare history have disappeared from my own history.
Any older notification of product purchases from Apple may have included an email, with serial or other number identifier. If you saved sales receipt or invoice, etc. Some products had the serial numbers on outside of case, (or inside on circuit board) along with model or sales numbers.
Sorry to not have an exact answer. You may have had some ability to restore the missing information if Apple service had been contacted sooner; not sure how long or where this kind of info may have been retained at a service facility. Perhaps an authorized service can replace the missing SN with another, to help restore lost functions.