Variations are supported across all channels that support them. Listings in Listing Mirror may either use the same variations across different channels, or define them differently on each channel.
At the core of Listing Mirror, all items are tracked at the SKU level. So a T-shirt in Red and a T-Shirt in Blue would have different SKUs, different Quantities, and various other attributes. They could be bundled together on Amazon as a variation, but shown on eBay as two different listings.
For a specific walk-through of how to use Variations in Listing Mirror, look at Creating eBay Variation Listings in Listing Mirror . It is very similar across marketplaces to set up variations.