How to get unique FNSKU rather than the ASIN?

If your product is already listed then while converting the listing from Amazon FBM to FBA, kindly select the amazon barcodes to changes the FNSKU of the product.
But When you are creating the listing of your product, choose FBM fulfillment method and then list your product. After listing the product wait for few minutes and then convert your listing to FBA and while converting the listing from Amazon FBM to FBA, kindly select the amazon barcodes to changes the FNSKU of the product. In this way your FNSKU will be changed.

If your product is already listed then while converting to FBA select Amazon barcodes. If you want to create a new listing then first create the listing as FBM and once the listing is created then convert it to FBA by using the method described above.

If you want to change the FNSKU for your listing then you need to convert your listing to FBM first and then select amazon barcodes for getting a unique FNSKU.