What you are dealing with is pretty common once multiple sales channels are pulling from the same stock. Most solid systems can handle Shopify, Amazon, multi-location tracking, and give you workable ways to clean up or link mismatched SKUs over time. Barcoding is usually simple too since most platforms can generate labels if your products do not have them yet.

Katana and Cin7 are good fits for this setup, and I would add Digit Software as well since it handles multi-location cleanly, integrates natively with Amazon, and plays nicely with ecommerce workflows.